I'm pretty much the same size except for the weight, I do have a couple of back issues and sciatica. The pain comes and goes, sitting, standing too long, and walking (sometimes) triggers it. The one thing that doesn't ever cause pain, and in fact makes it go away, is riding my drop bar bikes. Of course what works for me likely won't work the same for you.
Maybe you can rent or borrow a couple of different bikes to make sure you know will be comfortable for you? Lots of good choices with your budget, the position/fit is the most important thing to be sure of first. The hand and wrist position
MRT2 mentions is a real issue. My wife was getting serious shoulder pain (to the point of doctors, PT and cortisone injections) before changing from a flat bar to drop bar bike. Again plenty of people ride flat bar bikes with no issues, but it is something to investigate. Good luck.