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I would get a Thinkpad, either used or new, with that budget. Generally all the hardware will work out of the box, with the possible exception of the fingerprint reader if it exists. RAM and SSD should be replaceable, so if you purchase new just do the upgrade yourself to save some bucks.
The only time I’ve willingly used a banking app is when they lock out my VPN IP. In those cases the app still usually works for whatever reason. So far, I haven’t found any functionality missing from the webapp, but I’m also dealing with brick and mortar institutions.
I would be concerned as they will eventually probably move to a phone app first ecosystem, however it will probably take a while. Some people are still only using in person and phone banking (where you call them and punch in numbers).