Im in a similar situation. The .270 is a great caliber but it lacks the real heavy for caliber bullets. I actually found some 165 and 175 VLD grainers in .277 but you'd need a faster twist to use them. I'm sort of stuck between the 6.5mm and 7mm. The .270 win and 280 rem are slightly more efficient than the 6.5 resulting in more muzzle energy, which makes me lean towards the 7mm. Heavier bullets than .270 with higher BC's and SD's makes it desirable. the 6.5 will most likely shoot flatter because of the higher muzzle velocity, due to lighter bullets. There's also the option of taking it one step further and going to the .264 win mag. Im not sure how easily an 06' action can be converted to use the .264 win mag. AFAIK the case length is the same but the magnum has a slightly wider base.
I think either way you're going to have a nicer shooter. If I had the choice between the two you listed I'd probably choose the 6.5mm with the longest barrel that I felt comfortable with. The only reason i'd go for the .270 is because the availability of factory ammo over the 6.5mm