Speaking from personal experience stringing in a retail shop for several years:
I am happy to string with any viable string provided by the customer. And as is the case on any string job, I will stand behind my work. That is, in the event that a customer brings their racquet back with a complaint, we will assess if we made a mistake which caused their dissatisfaction. If so, we will work with the customer to make the situation right. However, I will not warranty string provided by the customer. Though it is not common in my experience, string can be defective or compromised in some way, which could cause the string to break prematurely. I have no idea where your string came from, how old it is, where it was stored, etc. The bottom line is, I’m not going to warranty any product that doesn’t come out of my inventory.
The above, by the way, is not all that dissimilar from your local car mechanic - if your mechanic will allow you to provide your own parts for work he’s going to do on your car (and that is a big if), no legit mechanic is going to stand behind parts you provided - if they break, don’t last, something is wrong with them, etc, that’s between you and the parts mfg/distributor, not your mechanic.
Now, in your scenario OP...did the shop have TGV in stock? Or did you provide them with a set of TGV to put in your racquet? If it’s the former, the shop should have eaten the broken set of string and restrung your racquet, no questions asked. For me, you shouldn’t have had to pay anything extra - other than for the 1 set of string and 1 labor charge.
If however you provided your own string, and it broke during stringing...for me you’re SOL, as so far as the initial set of string you provided. Either you’ll have to provide another set of string and I’ll try again, or tell me what string I have in inventory you’d like instead. Some shops in this scenario might even try to charge you labor twice - once for the 1st string job where the string broke, and another for the subsequent job. Me, I think that’s tacky - I’d only charge you labor for a completed string job.