I can get pain in the first MCP joint for two reasons.

One is autoimmune inflammatory arthritis for four to six months after a flu shot (stopped getting them several years ago for that reason) - that's pretty distinct with redness, swelling and heat.

The other is directly connected to my shoulder injury. Mostly the subscapularis, but the whole shoulder girdle is connected to the hand. The fascial line pulls the thumb out of alignment and wears on the joint. I know you've been working super hard on good shoulder alignment - the things my yoga teacher emphasizes really help me here - warm up with full hand stretch with arm extended either overhead or to the side, focusing on stretching each finger away from the thumb sequentially, then the whole hand as wide as possible, holding for a good 30 seconds in each position. Good hand positioning when we're weight-bearing on the hands doesn't seem to affect my thumb joints as much, but it's so critical to shoulder alignment that it bears repeating - weight concentrated on the second and third metacarpal heads, none whatever on the pisiform, feeling as though you're sucking the center palm in and up, wrist crease perpendicular to the body. No clue about kettlebell grip, but if there's anything you can do to keep your hand bones in good alignment with each other and with your shoulder there, I'd be conscious of it. If anything's going to help rebuild the joint space, keeping the fascia healthy and creating some space for the body to rebuild cartilage is a prerequisite.

You probably already take glucosamine - it does help my knees - what I've seen in the literature favors glucosamine sulfate over hydrochloride, and warns that rebuilding joint space doesn't necessarily relieve pain - but I think it's worth a try if you haven't already.

Hope you can get some relief.