Not sure what "break the bank" is for each person, as it seems some are comfortable with $500 rifles, while others have no issue with $5000 rifles. I'm personally somewhere in between.
My gun buying has probably increased substantially since I retired, as we don't owe anyone really. So I've purchased some rifles I never thought I'd ever own the last 11 years since I retired.
But my choice would be a single shot like a Rolling Block, and caliber would be .40-65 Win. as it can allow for a variety of bullet weights, powder charges, and vast distance choices should you choose to shoot it anything from 100 yds. to 1000 yds.
Most are heavy enough to handle recoil well, even with heavy 420 gr. bullets, but with the modern twist rates most .40 caliber barrels have of 1:16" twist, these guns shoot well with anything from 300-420 gr. bullets.
My Rolling Block Sporting Rifle in .40-65 is a joy to shoot with almost any bullet weight at around 1350-1400 fps., and isn't tough on my 71 yr. old bones at all. Most of these guns can be found for anywhere from just under $1,000 to maybe just over $2,000 if you went crazy with options, or bought original Remington rifles.
This one ran a little over $1200 by the time I had it all setup with sights. 34" octagon barrel, and weighs about 12 lbs.