ok hear me out:

what if you *didn’t* have to sign in to collect

I'm not familiar with the service in the screenshot, so this might be way off base (HA! Get it, base). 

One fun thing with frames is using a normal POST button, then picking an address from the verified payload. Address in hand, use whatever method you prefer to cover the gas for the user, and mint a token to them.

Website equiv would be spinning up a custodial wallet and minting to that, or have the user sign a message with their wallet and mint to that, covering the gas using whatever method you like.

Sign in is a nice gate to slow down common bots from burning through your gas subsidies, though.