That is one odd-looking setup. It looks like an old institutional
sink. I would try everything before ripping it out, including
contacting the maker of the faucet, and getting another plumber out to
look.

Ditto the timeframe to repair everything, it will take a couple weeks
of trial and error and trips to the hardware store.

If it turns out to be impossible to repair the faucet I suggest just
removing the sink and getting a omdern one that takes modern fixtures,
then fill in tile around it, whatever looks more or less OK, and let it
go at that. Losing your only bathroom is really painful.