1. Heat shrink sleeve. You remove and throw away your existing grip, possibly remove the staple, clean up the handle, slide on the black heat shrink sleeve, take a heat gun, gradually heat it up from throat towards the end of handle while rotating, let it cool, cut it with knife at the butt cap and where the grip should end (before heating up, it should overlap by say 3-4 mm over the final length by the throat and more so on the butt cap side), install new grip. You can leave the staple out. About $15 in materials, about half an hour of work if you are not a pro, and you need a couple of basic tools, notably heat gun.
If you have not done it before, there are videos on youtube which show how to do it. It is very easy.
2. Overgrip. Just install it on top of the grip, about $3 and about 5 mins, the only required tools are pen and scissors. It may change your grip size somewhat, but less than a size (1/8). Maybe by about 1/16 - which is half size.
3. New grip. To the best of my understanding, it will not change your grip size at all unless you find something totally unusual and nonstandard, like very thick. They are all padded (i.e., quite a bit thicker than overgrip) and all of similar thickness, take or give a small +/-. Replacing it will refresh your racquet at a cost of $8 - $15, but will not do anything for the grip size.
Hence, you can meet your goal of increasing grip size to the next size ONLY with heat shrink sleeve. Note that heat shrink sleeves come in two difference sizes, 1/8 (full size, black) and 1/16 (half size, white).
Bevel feel will remain after grip enlargement, but will get slightly muted, i.e., transitions from bevel to bevel will get slightly more rounded.