I'm obviously missing something here. If your trailer has a broken axel how are you going to tow the trailer or trailer and boat anywhere? Second I don't understand why the repair shop wants the boat off the trailer or why they would charge you an additional $200 for the boat being on the trailer. There is no reason to take the boat to off the trailer to put a new axel on. And what does the shop having a car in "resto" have to do with you bringing in your trailer or boat and trailer togethper.
They should be able to put a new axel on your trailer with the boat on in a couple hours. An easy quick job for a pro.
Finally I live in California And I don't know off hand where you could consider anchoring a boat in a river unattended for a couple days other than one of the rivers that are a part of the Delta system like the Sacramento River, San Joaquin river , or one of the others. In any event I wouldn't leave a boat unattended as too many things can happen. It could be vandalized, it could be swept away by current, tide, wind. It also may be considered a navigation hazard especially when anchored over night and no visible lights on it. As far as a public lake most in California are patrolled by county sheriffs dept. they will notice if a boat doesn't move for a day or two and will probably tow the boat and slap a fine on you.
This shop sounds pretty funky. Find a more reputable professional shop. Once the have the parts they can get you in and out in a couple hours.