Based on the information in your post, and particularly with the fact you talked to your PT and he gave the green light, there is nothing to prevent you from training at the gym. The challenge is figuring out what you can do. In this answer I'm going with the presumption that the balance problems has to do with strength and muscular stability rather than an inner ear vertigo type problem.
If you can't stand, you can't squat or deadlift, so there goes some of the best lower body work options. I'd recommend no more than 2x a week when just starting out. When you feel confident with that, you can move on to a more typical 3x a week plan. The plan I'm laying out here is a relatively simple one, and just a starting place:
- Leg press 3x5
- Leg curl 3x5
- Glute bridge 3x5
- Incline press (dumbbell or barbell) 3x5
- Curls 3x8-12
Just start at the lightest setting on the leg press and leg curl machines. Glute bridges should be done body weight only. If you can't do them just yet, work on what you can until you build up enough strength for the glute bridges. If you can, hold at the top for a second on the glute bridges. Same with the incline and curls--start light and build up.
If you hit all 3x5, you can add 5 lbs. If not, just stay at that weight until you can. The reason for starting at the lightest settings or weight is to build up momentum. There's probably a lot of stabilizer muscles that haven't really been used as much, so the lighter weight is going to target them first. The more muscle you have in your legs and glutes the easier it is to stand.
Build from there
When you feel like it's possible, you'll be making some alterations to the basic plan:
- Leg press 3x5
- Back extensions 3x8-12
- Alternate: bench press/seated overhead press 3x5
- Curls: 3x8-12
You can stick with the incline if you want, but this will give you a bit more variation. The back extensions work a bit more of your posterior chain than the glute bridges and leg curls.
By the time you can leg press your body weight for reps, you should be a lot more stable than you used to be. Ideally, you would build up to being able to do goblet squats, and then barbells squats up to your body weight. However, your journey is likely to be different. You may have to make some changes to the ideas I put up for you based on ability. If your balance issue is more of an inner ear problem rather than strength, you may never be able to squat--but there are things you can do.