I might as well toss my opinion/2 cents worth into the fray.
I have several guns in the class you want. 243 I do not have. however, I have two calibers that no one has even mentioned yet. 25-06, that is a good varminter and it will take down a deer with 120 grain slugs at over the 600 yard max you have said. you also could look into the 257 Roberts, I own one of these too and its no slouch at all. with 90 grain sierra HPBT's I can sink just about any deer I know of. with 65 grain V-max you could explode most varmints or even Coyotes. My 257 is the go to rifle when I want to take on coyotes at 300 yards or longer. nothing says "gotcha" like a 90 grain slug that I can drive tacks with at 300+ yards. I know you can not find much variety of ammo.. I reload so I can make my own tailored to my application. 75 to 100 grainers for deer, light 60 grain slugs for coyotes, and if I get a varmint in my sights.. they have a hard time standing up to a RNFP slug designed for a 25-35 or 25-20.. red mist time. its been a long time since I worked the bolt on my 25-06. I am not even sure why I have not pulled out the old BDL heavy barrel.
one of my customers just picked a bullbarreled, 25-06 Rem BDL for 200 dollars out of a pawn shop. it seemed no one wanted the caliber. I inspected it. maybe 20 rounds down range. it was a Varmint Special, very sweet trigger, and it came with a burris, Black Diamond 6-24X50MM scope mounted to it. the scope is worth over 700 dollars alone. the rifle was worth at least 800 dollars. all that for 295 out the door. look around.. you might be able to find something like this for really cheap. I am also told, since I live in antelope territory, the 25-06 is a great antelope gun too.