Does look like you'd be safe to increase your basal on those nights. That'd be the first thing I'd try.
Actually no scratch that. First thing I'd do would be to also basal test on a non-weight-lifting night. I notice that most of the rise happens between 4 a.m. and 6:30, so after your 1u correction at midnight is done working, and right in the period of time where most people tend to have dawn phenomenon. You might also be having dawn phenomenon on regular nights, only you don't notice it because you don't have the additional rise from the weight lifting. Going to sleep and waking up with the same BG level tells you very little about what's actually happening to your BG overnight. I realize this doesn't seem totally relevant to figuring out what to do on your weight-lifting nights, but knowing what your nighttime pattern normally looks like can help you to see by contrast just what the weight lifting is actually doing to your BG.
For info: it's not an accurate basal test if you also have fast-acting insulin on board at the time. But you weren't really trying to basal test here, just to see whether you were dipping or rising at night. Since you did still see a rise, you can still confirm that you're safe to increase your basal on weight-lifting nights. But you don't actually know just how much of a rise you actually get nor when exactly it happens (what the curve is) because that bolus insulin can't be ruled out. And had you had a dip instead, it would have been impossible to draw any conclusions. It's ok to let your BG run a bit higher during a basal test so long as it isn't dangerously high (say, over 15 or wherever you theoretically test for ketones) in order to get clear data.
For further info: if 1u drops you by 5 mmol/L, correcting a 9.3 could theoretically have brought you down to 4ish overnight, yikes! Since you are so sensitive, have you considered getting half-unit pens? (Although no pen can accurately dose out less than 2 "clicks", so on a full-unit pen, the smallest accurately delivered dose is 2u, and 1u for a half-unit pen. Less than that and you can't be certain that you got the entirety of the dose due to the mechanics of the plunger).
If increasing basal doesn't work, it seems counterintuitive but I might try having a small amount of slow carbs with dinner or at bedtime. Or even just after the workout. How the body reacts after exercise has a lot to do with the fact that it is replenishing its stores and rebuilding tissue/repairing microtears. Sometimes trying to do that on too little fuel can lead the brain to declare a state of emergency, which causes some protective mechanisms to kick in that naturally increase BG and/or IR, so you can end up with higher BG from lower carbs and lower BG from higher carbs. Doesn't always apply of course, but when you're stumped, it's worth trying everything!
Another possibility could be to split your workout sessions - not meaning to pry, you may already be doing this; ignore if so! But say if you're doing a full-body session every other day, switch to doing lower body one day, arms and core the next, or however you care to split it up, but so that you're doing less intense sessions more often (less stressful to the body). And/or, add a cardio workout on your weightlifting days, a small one so you don't exhaust yourself, but it might help to counteract the weightlifting rise.