7mm08 and 308 are similar enough that you will probably find more difference from other factors like weight of rifle, weight of projectile choice, shape and fit of rifle.
For what its worth I have a ~7lb (+suppressor) 308 and a ~7.5lb unsuppressed 7mm08, subjectively the recoil feels very similar between the suppressed 308 and unsuppressed 7mm08. 150gr in the 308 but used to use 125gr, and currently 120gr in the 7mm08.
If you want to manage recoil better then don't worry about buying the lightest fashionable plastic rifle , get something a bit heavier that fits you well.
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