Squat, Lunge, Deadlift with mobility band to break your pattern.

The amount of tension your mobility should maintain in order to promote you to shift weight into your habitually limited side will throw you off balance.  If done correctly, the only place to perform this safely and effectively are at the Smith machine (except for the less seen squat/lunge apparatus with shoulder pads that leave the back of you open).

The correction the mobility band provides cannot be executed properly without the use of the band.  You will notice your resistance will need to be reduced, that you may need one or both arms to help maintain form, that muscles you do not typically engage will engage, and that you are more sore (despite reduced resistance) in areas you are not usually sore.

The following videos explain the form to promote to challenge your old pattern and the pull of the mobility band.

Lunges using this particular form -- not to say you can't do the way you want in addition, but this is the way to attempt to strengthen out of the way your body prefers to drift.

For squat, but also dead lift.

Adjusting how you work out helps prevent the things we have corrected together from eventually returning.  If you always lapse into your "flexible" hip capsule, add weight, and add repetitions, you not only bring back your old shifts and rotations of the body, but also strengthen into that pattern.  This is a way to break that pattern while bringing 'stretch' to the limited side and then building strength into a more symmetrical pattern of motion.

Best physical therapy practice is whole body, lifestyle changes, how to improve movement patterns, but also how to tailor it to fit what's going on specifically with YOU.