Bullet placement is paramount. As has already been said, either round will work just fine. I would shoot the one you shoot the best, and if that is a toss-up, assuming you can get more than one tag, blood them both!
Now we're talkin!

Thanks for all the thoughts, seems like a real toss up but I'm glad to know how others with personal exposure to handgun deer ventilation view these options.
I'm very comfortable with the 45 and always leave the range grinning about its accuracy. The 19 is new to me, remains to be seen. I'm pretty good with all my Smith and Wesson wheel guns though and generally feel quickest on the sights with L frames.
I'll put in some regular practice with both for a while. I feel free to choose either way now, which is great.