Would it be possible to create a filter which would put the logos of the largest donors on every congressman’s clothing, when they appeared on TV?

Technically, maybe, if the critter isn't moving around much. It would require a live connection to a database where the associations are stored (and kept up to date). Much as I hate to say it, some AI might be helpful here. The catch, of course, is that the news station doing the broadcast would have to cooperate in applying the overlay, so you would probably only see it done on Republicans.

Kind of like how the billboard-y ads on sports programs that go over the green-screen backgrounds change?

I like it! I don't know what prep would have to be done - maybe make every Congresscritter wear a green jacket?