Any of the cartridges you mentioned will work for prairie dogs. I've heard of people shooting them with .300 Winchester Magnums. As someone else pointed out, the volume of shooting done while going after prairie dogs is the major concern. If you can handle several hundred rounds from the bench with the aforementioned cartridges, then by all means, go ahead with it. Of the ones you mentioned, I've only shot the .243 Winchester. It kicks significantly less than the .308 Winchester. I can handle, at most, 80 rounds of .308 Winchester from the bench in a day's worth of shooting (paper targets, not p-dogs). I could handle quite a bit more from the .243 Winchester.
I don't know about the .25 calibers or the .260 you mentioned, as I've never shot any of them.
I'm going to suggest something off the wall here:
How about a T/C Contender or Encore rifle? You can set either up for varminting with barrels in .22 Hornet, .222, .223, .221 Fireball, and .218 Bee, to name a few. The Encore can also handle .22-250, .220 Swift, .225 Winchester, and .22 Savage Hi-Power. For deer, either can handle 6.8 SPC, 7-30 Waters, .30-30 Winchester, and quite a few other cartridges. The Encore can also handle all of the calibers you mentioned, as well as 7mm Mag and a whole host of other calibers suitable for game both larger and smaller than deer.
A new barrel for a T/C Contender or Encore is going to cost less than all but the most inexpensive of bolt-action rifles. If you're going to buy another gun, why not make it as versatile as possible?