I was thinking of getting my hands on a used 7MM Rem Mag, either a Rem 700 or Ruger 77 and having a fluted ER Shaw barrel and a Synthetic stock it for a lighter weight packing rifle. Should be too expensive. Any thoughts? I have never owned a 7mm Rem, I did have a 300 Win Mag which I did not care for

Just get a tikka

Yes,especially the Superlight version.
I expect the 7mag at 7 pounds scoped will be about as light enough for anything close to tolerable recoil.

The Ruger isn't the way to lighter weight. Inexpensive synthetics are generally on the heavier end too.

And no one makes a quality lightweight synthetic for a Ruger. I'll 2nd just buying a Tikka.