Sorry, this will be long but I've been meaning to post a lot of this for a while in a searchable place case it helps someone with similar problems.
I started playing tennis consistently about 4 years ago; almost all on hard courts. In no time at all I had significant foot pain: like someone driving nails through the tops of my feet down through the arch.
My experience says that there are no shoes made that stock provide enough or the right arch support for me to be comfortable. Once you start going into orthotics, the arch support implicit in the shoe is almost a non-factor...in many cases this is based on the shape of the insole to a large degree and you may well be discarding that anyway. So the main thing may be to figure out shoes that work well for the inserts you end up using...the arch support is more down to those inserts than the build of the shoe. [Caveat: this is my subjective experience, I wouldn't want anyone to think I'm representing myself as an expert!]
Long story short I now swear by custom orthotics as prescribed by my podiatrist (I found one locally who's a runner so he's particularly interested in helping/fixing other athletes). I use Mi-Adidas Barricade V's (wide fit!

) to house the orthotics. Expensive up front, but now I play without pain several times a week...easily worth it in my view.
I started with off the shelf arch supports (short ones, plastic "shell", place under the regular insole). Helped a bit but I still had significant pain. From there I tried SuperFeet Green (replace stock insole). Those helped more but I guess I am tough on feet as I pound the court -- I more or less destroyed the forefoot of these within a couple of weeks and still had some pain. Enter the custom fit orthotics -- add 4 weeks elapsed time (still playing regularly though) and no more pain.
[ one aside about orthotics: the first set I got prescribed were very wide, especially in the heel area. The Doc had to grind them to get them to sit in even NB shoes which have more than average heel area to work with compared to most every other tennis shoe I've tried. Net result: we returned the first type of othotic and went with one that was narrower in design -- it's nominally for use in dress shoes I understand but it is narrower throughout its length and this makes *all* the difference in the number of tennis shoes that the orthotic will sit in at all much less well. ]
I tried a lot of different shoes trying to find ones that work with orthotics. Not all do by any means.
And the other thing about shoes that no one seems to talk about is the shape of the last (think: plan-form template/outline) they are made on. My feet are wide in terms of edge to edge measurement but they are also rather straight. Most folks have feet with a sort of "C" curve to them -- consider your left foot -- heal and big toe along a straight line, your midfoot sits pushed out a bit to the left of that line...now draw a line from center of heel following the centerline of the foot and you'd see that line has a bowed curve to it. In my case, the arch area is both flat in height terms and not pushed out to the side relative so the midline of my foot is rather straight, not bowed so much. That all means I have to use shoes that are on a last where the midline is also somewhat straight or my outer toes end up getting crushed.
So...in general terms what I learned about shoes for me and my odd feet and orthotic arch supports is:
New Balance: straight-ish lasts, flat and wide foot bed below the stock liner which makes orthotics fit in them well. CT1200WB in 2E worked rather well for me...but they stopped making them or they got hard to find at least. The 1004's were poor by comparison mostly because for me I didn't find them stable enough when running fast and changing direction a lot.
Asics: I've had good luck with Asics for track before. For tennis Gel Resolutions were comfy and fit the orthotics quite well -- flat foot bed and wide enough in the heel cup. These didn't last well for me though (uppers coming apart a bit) and they didn't feel stable enough for me with fast direction changes when playing.
Prince: tried OC-1's in wide and Scream QTs. Ok for using orthotics in those. My main issue here was too much "squish" I think. In general I wasn't able to feel the court well when wearing Princes making me slower around the place. I started to feel some heel pain with both types so oddly enough I guess I have to conclude that more padding is actually a bad thing for me -- counter-intuitive perhaps but less padded shoes just seem to feel and work better for me anyway.
Reebok: tried a couple of different types. Good compatibility with the orthotics (wide/flat foot bed, removable insole) but these were limited to being good in straight lines only and they didn't last all that long (remember though: I believe I wear hard on tennis shoes relative to the average of other players I see around me). In essence I felt like I was sliding off the side of the foot bed too much in the Reeboks I tried.
Wilson: very poor fit for orthotics because the heel cups seemed so narrow. At least one variety had the insole glued in too -- what a mess that made. Pretty good space for width in the toe box though.
Babolat: wrong shaped last for me it seems. Pretty darned narrow all the way through which made fit a problem and getting orthotics to sit in them. I hear propulse2 is a little wider but I haven't tried that one.
K-Swiss: at least one retailer told me this is the go-to shoe for wide/flat feet. Indeed, flat/wide foot bed made the orthotics work well in them. However, my heel felt low relative to the toe area compared to most other shoes. I didn't really think about this much until after a few sets and I started to get knee pain which was a new thing for me. Stopped using the K-Swiss and the knee pain abated. This was the Stabilator, by the way...others may be different I guess but given earlier experiences stability orientation seemed like something I needed. Pity because they were nice light shoes and I felt "fast" in them...but more pain...well couldn't sign up for that.
Adidas: most all of the Adidas shoes are very narrow and quite C shaped in the last too. I tried the Prevails on the basis that they were reviewed as nice and wide -- perhaps true in edge to edge dimension but the swoop/curve on the outside of the forefoot was so severe I could barely get my toes in there...not going to work. I tried Barricade IV's more or less on a lark and they were too narrow on the outside of the forefoot really but closer to OK than I expected. Barricade V's came out reviewed a bit wider and these were pretty good but only after a month or so...in retrospect I think now that this was normal extended break-in *plus* some time to wedge the toe box a little wider with wear. With respect to the orthotics, the Adidas shoes are barely wide enough in the heel area and the foot bed there is not flat either. What I've discovered though is that the width to fit the orthotics in, letting them sit down in the shoe so my heel is held rather than floating out of the show when I run hard, is the key... Over time the orthotics "bed" themselves down inside the Barricades -- they sort of cut into the curve of the foot bed to make a flat landing zone for themselves (did I mention that I pound the ground pretty hard??

) in a week or two -- looks a mess but doesn't harm the integrity or longevity of the shoes fortunately.
Then I learned about Mi-Adidas and the fact that they will make you Barricades with a wide fitting. Can't get that at retail. These are still the reigning champs for me and I'm on my second set now having worn the first ones out (tread wear at around 8 months of play so outside the warranty period). In a wide fitting they feel comfy on out of the box to walk in and I'd say it takes a week or so for them to get that carpet-slipper kind of comfy. I guess this is closer to the experience folks with normal feet have with stock Barricades with respect to break in.
I'm still in search of the perfect shoe since the Barricade V's are good but I'd prefer them wider and less C shaped in last in the standard fitting. Not to mention people are saying the VI's seem to be more narrow again -- oh no!!! But for now, Mi-Adidas Barricade V's with my custom orthotics let me play consistently without pain from my feet.
I suspect a lot of the above is subjective and specific to my own situation of course. I had a lot of help from a podiatrist MD who's an athlete himself and I suspect that anyone with foot problems would similarly benefit from that sort of help.
However, the lengthy stuff above may help someone if not the OP and I can't think of a better place to post it than TW where it might be searched up by someone with the same issues I started with. Hopefully all that investment in shoes that I have made might help someone at least!!
Bottom line though: persistence paid off for me -- so take heart, it can be
done!