I would say that it is highly unlikely that patching the non-dominant eye will help you with your binocular vision, since if you have trouble with your binocular vision it can have many reason, but it is pretty rare that a weak or less engaged dominant eye is the cause. If you have a weak eye it is most likely your non-dominant eye so patching the dominant could help.
However, most of the time problems with binocular vision are due to the effect that both eyes could not work properly together which is best trained with both eyes open. @martin described some possibilities her which is also difficult for adults. Binocular vision issues can be caused by so many reasons that even hip, jaw, shoulder imbalances can play a huge role.
For most people who cover one eye it is most comfortable to cover their non-dominant eye, since you don't have to adapt when patching and when taken the patch of. Maybe when patching regularly you eyes are more relaxed than now and you can get some minutes/hours extra with both eyes before that become fatigued again.
I would try patching both eyes and give it some time and learn whats best for yourself even if that sounds boring.