I finally got it to work. Ive been spending a month on this trying to see if it is compatible. I was able to finally get the TM-U220 to show up on the config. Firstly I had an extra router that I connected to my computer, to the access point for internet and to the printer itself. I then changed the IP address from 192.168.192.168 to 192.168.192.167. Turned on the square register with WiFi on and it picked it up.
This is likely how square determines what driver/config to use so if TM-88 is discoverable, but doesn't print you may need to try another printer name using the compatibility list to see if your printer is close enough to an approved printer to actually print. I don't see any in a similar family name on the list, but if you do some trial and error you may be able to get that printer going. I would start with the TM-T88's.
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I found the solution. It seems that with every change of network configuration, even with a new computer start, turning the printer off and on again etc., the printer may request/receive a new IP from the router.