Broken auto pilots suck big time. I wanted to purchase an emergency helm mounted system so I could at least motor for a few hours and get some sleep. We had the auto break on rhe way to Charleston one afternoon and hand steered in 1 hour watches because it was so taxing staring at the compass at night with an overcast sky. Sucks!
So it's old tired men in plastic boats .
I feel the same way as you with hand steering and would probably do exactly what you're doing as well. Hopefully there's a technician to get it working quickly. It is all downwind and you could bite the bullett and just go. Perhaps If you talked to B&G they could close in on the problem and Danny could bring a part (if needed) to the T&C. That's an easy ride from where you are.
On the other hand,,,, I'd probably do as you're doing, staying as long as you need to get it repaired is a good plan. There's not much until you get to Nassau and there's not much there either.
I just spent the past 4 days in Annapolis fixing stuff the guy that winterized the boat neglected to do. I had broken pipes, leaks and an exploded filter housing when I returned and put her in the water.
Damn, good luck. Ah, the wonderful cruising boating life!