If the axles are original, they will need to be replaced on a trailer that old. If they have been replaced, ask for documentation.
The gutted bathroom may be a blessing in disguise since that era of trailer was prone to what is called "rear end separation. " This is where corrosion leads to the separation of the body from the frame. The fix involves opening up the back end from the inside and often replacing rotted sections of plywood subfloor. You can test for rear end separation by standing on the bumper. If the body doesn't move with the bumper when you bounce up and down, the frame and body have likely separated. Search the forums for "rear end separation" to see what sort of repairs it involves. We did our own repairs and ended up with a bathroom that I like better than the original.
Don't take the owner's word for what works. Who knows what might have happened since the last time they checked it.
Other than those slightly gloomy bits of advice, you're in for a wonderful adventure. And, if everything does actually work, that price seems reasonable to me.