If the G19 feels good in your hands I'd get the that - you could also try the G17 and see if the added weight helps.
I'm not Glock fan-boy (I don't even own one), but Glocks definitely have a unique grip, and so if the Glock feels good in your hands I'd try to stick with that.
Making your arms stronger is generally easier than finding a gun that feels good in the hand. Also, I'd think being stronger would be advantageous across the board for your profession.
If it's recoil that's fatiguing you - rather than holding the gun at arms length - it's probably your forearms that need strengthening. You could get one of those spring grippy things and work your forearms when you have down time. You can also get a ball of beeswax and work that with your hands.
If it's the holing a heavier gun (a loaded g19 is going to be heavier than most .22 pistols) at arms length, I'd agree with the pushups.
You said that it's okay for the first couple shots, so you're most of the way there. A few pushups before breakfast, and some forearm work in the car on your way to work, and you'll probably be all set.
si vis pacem para bellum