I would look at used phones on Swappa. Not sure anything is going to reach that pricepoint at new, and good is a big ask if you're trying to get under $100. I did a quick check on Amazon, this phonewouldn't be too horrible.
The Moto E4 would be a good choice. The unlocked/TMO version supports T-Mobile's VoLTE, has good reviews, and MetroPCS offers it for $60 after mail-in rebate.
If you don't feel like doing a MIR and want a fingerprint reader as well, it's also available on Amazon for $120: https://www.amazon.com/dp/B077Y79RBY
Yeah, that'd be amazing, actually. Especially since most smartphones nowadays (even relatively affordable <$400 ones like the Moto E4) would be able to run either of these games at the smoothest framerates possible.
If not a remaster, a ground-up remake with Fallout Shelter's art direction would be a nice alternative, too. The only problem Bethesda faces would have to be sales and marketing. Do they want to sell this game to hardcore/novice Fallout fans or would they rather tap into a more casual market instead?
Clearly, no one wants to see these remasters flop, so in the end, it's up to Bethesda to decide on how they're going to approach this.
At that price, you may want to check out swappa.com and see what preowned Sprint phones they have available.
https://www.amazon.com/Moto-4th-Generation-Unlocked-Exclusive/dp/B077Y79RBY/ This one you can use as BYOD
If you really want something new, this is another option. But it's much slower than the G5.
Moto E (4th Generation) - 16 GB - Unlocked (AT&T/Sprint/T-Mobile/Verizon) - Black - Prime Exclusive https://www.amazon.com/dp/B077Y79RBY/