I hardly doubt anyone can tell the difference in weight from a short to long action when holding the same rifle models with same barrel lengths. Only when you compare and see a few added ounces on paper do people start to say it's a disadvantage or drawback. The weight is closer to the shooter, so it's not like adding 2" to the end of the barrel where it's more noticeable. Keeping an eye on your scope weight fixes that issue if it is noticeable since that's where the weight is anyways.
In truth, the 270 is a better preforming hunting cartridge than the 6.5 Creedmoor. The Creedmoor is a great cartridge, but for "hunting applications" the 270 pull's ahead. Both will work but overall the 270 is better, less drop, more down range energy & is still light recoiling enough for average shooters. Here's a link from someone who did the math between the two --->
25-06 vs 6.5 Creedmoor vs 270
First time I shot a 270 I was amazed at how soft shooting it was, I expected it to be more due to the extra powder but that light 130 gr bullet doesn't offer much resistance going out the barrel. With copper/monolithic bullets out now the 130 gr bullets are capable of taking larger game so there's no reason to shoot 150 gr's anymore.
As per cost, the only time I see a higher cost is with magnum actions not long actions. I don't see ammo being any costlier for standard or premium hunting ammo between the 6.5 to 270. Only when you go into obscure rounds or magnum's do the prices go up.
Personally "if" i were to buy a 6.5 short action for hunting i'd most likely buy a 260 since it offers better performance with lighter weight bullets, it wasn't designed for the heavy target bullets to fit a short action magazine length like the Creedmoor so it holds more powder. I'm a fan of the 308 win cartridge design, I already reload for the 243, 308, & 338 fed, but I'd still choose the 270 if I could only have one rifle for the game the OP stated.