Lost Mode suspends payment cards and related services, so being asked to reauthenticate afterward is expected. (Apple Support¹)
Since the password prompt itself isn’t opening, I’d first make sure Lost Mode is completely off in Find My, then restart the iPhone and try Apple Cash again while connected to Wi-Fi or cellular. Apple also recommends checking that you’re still signed in to iCloud and that the Apple Cash account isn’t restricted or locked. (Apple Support²)
I wouldn’t sign out of your Apple Account or remove Apple Cash just to troubleshoot this. If the button continues to spin without presenting the password screen, I’d contact Apple Cash Support so they can check the account status.
[1]: Use Lost Mode in Find Devices on iCloud.com - Apple Support
[2]: If you can't send or receive money with Apple Cash - Apple Support