You could ask the present owner for a towing demonstration before you hand over the check. This would let you know if the trailer brakes do their job and the outside lights to include turn signal and brake lights are operational. Is your outside umbilical cord receptacle on your tow vehicle a match for the hook up on the trailer (some are 7 and some 8 pin)? The 1963
had copper water lines and no inline pressure reduction device, so it would be helpful to know that there are no leaks in the water system. You would have to rewinterize it after checking this though. Checking the toilet and then dumping it would let you know that it is functioning properly and that the tank is not clogged up. Check the referigerator for proper operation and see it works properly on gas and electric. If there is a tv antenna, take a small tv with you and see if it works.
These are only a few things that come to mind in addition to your list. Am certain others will fill in items not mentioned thus far.
Bambi is a fine trailer. We pulled ours from Florida to Alaska and back behind my 1994 Jimmy (V6) and lived in it for 6 weeks. A great trip.
My two cents, Pat