That IS good news; weren't you afraid he wouldn't take to it?
What a great riposte to all those people who claim squatting will damage your knees!
Whenever you feel like it! Or...whenever you feel you are no longer making progress with 3x10-12. But seriously: if you want to try heavier, nothing stands in the way of deciding to add 10-20% and dropping the reps, and therefore...
It sounds like you are ready for a few weeks of 5x5. Increase the weight to the point where you can just manage 5 sets of 5 reps. If you don't quite make the last 1-2 reps of the last set, that's okay: it means you judged the load right. A few weeks of that and you should find your strength has increased when you go back to 3x10-12 for hypertrophy.
I'm sure others have more interesting ideas, but I'd say just alternate them. The bodyweight days could really help to stretch you out after the full-body lifting days, especially if you increase the load on lifting days.
Good luck with what you do!