I had a transverse fracture in the middle of my left humerus 13 weeks ago. Should I persist in trying to let it heal naturally, or have an operation?

It depends on angle. If it was a simple fracture and the bone was in good alignment, let it heal on its own. If it was significantly displaced or angulated, a surgical reduction may be needed. But it is now 13 wks out, it is certainly healed/sealed back. So, if xray shows good angle/alignment, you are set, especially if no pain and doing well. Follow up with your ortho for consult if need be. Good luck.
Probably get it fixe. If you're losing elbow or shoulder motion, you'd do better with it fixed.