At your level it will be different to when you are at a higher level.
Open gym is a fantastic way to get more hours and focus on what you need to work on. But this is better for higher level gymnasts. At your level you still need the watchful eye of a coach to make sure you aren't practicing things incorrectly and getting bad habits. If you get bad habits now on your more basic skills they will be almost impossible to break and it will slow your progress down rather than speed it up. When you become more experienced open gym will be safer as you will have a better understanding of what your body is required to do.
Rec classes, maybe depending what your gym is offering. If your gym has classes that are at your ability level, with good coaches who pay attention to detail. Then it may be the way to go. But check with your pre team coach first. They will know if the class is likely to help or likely to be a hinderance, as they will have a better idea of the coaches abilities. If the gym offers a class like tumbling, or a back handspring class then this could be good.
Again its probably a little early for private lessons. As you say they cost a lot. You will get a lot from a private lesson if you are struggling with a particular skill. But they can often be disappointing for the amount you pay.
Another option is to do some cross training. Something that would really help your gymnastics at this point would be to take a ballet class, and/or other types of dance class. Even a class in trampolining if your gym offers it.
Home is also a good place to supplement your training. But try not to do too much at home or you will also learn bad habits. Stretching at home is usually fine. And you could ask your coach to write down a condition program for you. If you work on your strength and flexibility at home, you will find everything comes more easily to you at the gym.
In the long run it may be best just to stick with the training hours set for you. Your program will be designed to increase your hours slowly but safely. Doing too many hours early on is a bad idea. You can burn out and stop enjoying your classes as much. You know you are doing the right amount when you wish you could do more. Your parents may get burnt out from all the driving and things, or they may worry you are spending too much time in the gym and neglecting other things and they may pull you out. Also if you increase your hours too quickly before your body is used to it, you can get injured more easily and then you may not be able to train at all.