All 3 of those will get the job done. IMO just shoot your 300 for all big game and buy yourself a varmit only gun. Since you don't reload, the 223 is going to be the most economical to shoot and generally have ammo that's easiest to find.
Mueller makes a pretty decent tactical scope for about 200 bucks. Not Leu but OK value. You may want to consider a higher power than 12x... maybe 16 or more. Way more dough though if you're staying with the better glass like Leu... hence my mention of the Mueller. If you're wanting to stay in the 10x or 12x range then there are a gob of good ones out there for the money... Sightron, Elite 3200, Leupold xI or II, etc.