The 6.5X55mm case has 6 or 7% more capacity than the .260's, even more in practice when both are loaded to standard COALs with heavy bullets, which sees them having to seated very deep in the .260 Rem using up quite a lot of powder capacity. So loaded up for reasonable pressures in modern actions, the 6.5 will give a bit more performance
The issue for many is what action length is available or wanted, the 6.5 requiring long. So sniper rifle / tactical rifle competitors will go for the .260 with the option of the many good short bolt throw designs around with detachable box magazines. If a bit more performance is needed, the .260AI gives another 100-150 fps depending on bullet weight.
You've got a long action to play with, so have a real choice. I was in this position myself and went for a .260 long-throat (really a 6.5-08) that allows 140s to be seated with their bases just below the neck-shoulder junction. This is for solely target shooting, and this throating places constraints on the use of lighter bullets for varmints and small deer if I were in that business.
Brass wise, you've got really good Lapua 6.5X55 off the shelf that needs minimum preparation, and it's strong and long-lived. There is an Ackley version too that was popular in F-Class in Europe for a while that isn't too far short of 6.5-284 performance. If you go for .260 Rem, the American brass isn't as good but you can neck Lapua or Norma .243Win cases up and trim them (or .308Win down). This has the downside that doing so usually creates a noticeable 'doughnut' at the case-shoulder junction, that may cause problems depending on how deep bullets are seated.
For purely target shootingf, I think I'd go with 6.5X55mm if I was making the choice again today for performance and brass preparation reasons. In fact, I've considered going back to the gunsmith to have the barrel rechambered.
You want a multi-purpose rifle though and that makes things trickier depending on the bullet weight(s) you want to use. The 6.5X55 and 6.5-08 throats are really designed for 140s, so 90-120s make a long jump into the rifling. If you're always going to use 130s and up, it's less of an issue. If you want to use the lighter stuff, I'd say go for .260 Rem and discuss the reamer with the gunsmith to come up with as good a compromise as you can depending on the mix of shooting.
Over here in the UK, in Scotland to be precise, we have a top sporting rifle builder (Callum Ferguson of Precision Rifle Services) who almost specialises in .260 Rem usually built on Borden actions. He throats the barrel 'short' so it's suited to varmint bullets, but will still handle the 100gn Nosler Partition which he says is more than adequate for any British deer species including Scottish red stags. He has plenty of satisfied customers with these rifles - so that's an alternative approach for a multi-purpose rig.
1-8.5" twist is the norm and handles all the usual sporting and match bullets; you can go for a little slower twist if you won't use the heavies.
Accuracy wise, I don't think there's anything between them if everything else is equal - the 6.5 has a reputation for superlative accuracy, but that was high quality Swedish military rifles and ammunition matched against often not so high quality military stuff from elsewhere. Put the pair in custom rifles and use equally good brass and bullets and you'll be hard pressed to tell them apart.
Laurie.