The letter you got was from a law firm in Florida that does nothing but send letter such as the one your daughter got.
If you pay, it means you have settled any civil claim for the $350 they want. It has absolutely nothing to do with any criminal prosecution. Paying it won't guarantee no criminal charges will be filed, nor will ignoring it cause criminal charges to be filed.
If she ignores the letter (and the second, third and fourth that are likely to follow, each sounding more and more urgent and threatening), then they must make a choice - file a civil lawsuit against her or let it go.
I have yet to speak with an attorney whose client has been sued in a situation like this. The law firm that sent your daughter this letter operates on volume. For the price of a form letter and a stamp, if they collect several hundred dollars, it's a huge windfall for them. Based simply on the volume of letters they send out, if even a fraction pay their outrageous demands, they collect large sums.
Let's say they actually do file a suit against your daughter. She can always choose to settle it later or fight it. If she fights it, the store would have to prove their "damages." The wallet was recovered and I presume was able to be resold. Damages = $0. The store personnel that were on duty were already on the clock, so presumably the store didn't incur any overtime costs above and beyond what they already pay their employees. Damages = $0. Even in the best case scenario for the store, let's say the court awarded the hourly wage for two store employees for two hours worth of time to deal with this situation. What do they make? $20 an hour each? Total damages (even in the best of all worlds) - $80... far less than the $350 this law firm is demanding.
If you pay in full, it goes away. If you offer to pay half (or some other lesser amount), it potentially opens the door to further harassment by them.