Going heavy on squats may do more harm there than the heavy pressing. How good do you think it is to put a heavy bar right on top of the disc(s) that are already injured?
You have to stop doing movements that cause the strain, or else the injured tissues will never heal.
Discs do not just randomly bulge for no reason. They bulge because of forces put on them. You have to remove/reduce the forces that cause that disc to bulge. You need to introduce new forces to squeeze it back to its ideal location (like thoracic extension).
So for example, if you're in grad school spending a lot of time hunched over a computer, the ligaments and small muscles connecting the vertebrae in the mid-back and neck get stretched away from their ideal length-tension relationships. The discs, which are viscoelastic, slowly deform. The firm jelly-like nucleus squishes out; hours and hours of pressure stress the annulus (fibrous outer container for the jelly) in that area. Then you get up, with the tissues still deformed (see McGill, Low Back Disorders), and try to lift heavy at the gym. Now you put high loads on those overstretched tissues, causing strain.
So the simple bottom line: you HAVE to eliminate the movements and postures that cause the problem so that the injury can heal. If you keep producing the forces that bulge the disc, it's not going to stop bulging.
You don't have enough thoracic extension or scapular retraction. So discs are always deformed out of their ideal position. If you can't extend as much as the girl in Cressey's video (Gentilcore extends more than that!), your muscles are currently unable to keep your thoracic spine in the ideal position, and also unable to stabilize your shoulder movements. The disc bulge and AC joint injury are related by this same underlying problem.
Nobody ever wants to hear this. But if your thoracic spine is locked into too much flexion and your scapulae are locked up anteriorly in unloaded postures, all loaded movements you do in the gym can cause further harm. Take rows for example. In theory, rows work the scap retractors and mid-back muscles, so they should be a good choice. BUT if your thoracic spine cannot extend, THEY DON'T. If you can't extend the thoracic spine over a foam roller, with gravity to help and partial spinal support in the lying position to make it easier, than how are you going to maintain safe thoracic spine positioning in loaded movements in the gym? You cannot. So even things like bent-over rows can put bulging forces on your discs.
One thing I have learned, the hard way, is that your LONG-TERM progress in bodybuilding is no better than your joints' condition right now, ESPECIALLY the spine.