M.G.
We had something similar - not at all to do with constipation or Meatal Stenosis. Having to pee right after going, and also a sudden urgency. No UTI and had them checked out. We were told have them take their time when they pee. Helped in our case. Sometimes little kids don't quite empty their bladders - either rushing or just don't recognize that there's still more there. Pediatricians will often suggest to kids who wet beds to go to the bathroom twice before bed. It's not always helpful, but it helps to empty the bladder fully if they haven't.
I've had friends go through this and it's usually a stage like ours was. Constipation can cause pressure there, and also little spasms, so it's entirely possible too. Eliminate the constipation and see if that helps, and try having him fully make sure he's fully emptied his bladder. See how that goes. If not, check with his doctor.
Good luck :)