I do not know how to salvage Savage attacker so it does not get overpowered in the process.
+1 damage to all attacks, +2 with heavy weapons?
Savage Attacker rerolling your weapon damage dice once per turn is a really silly feat because it slows down combat and provides minimal benefit. Moreover, it's a trap, because folks who don't do the math may not realize how weak it is.
The benefit is as follows, assuming you always reroll after a damage roll below the average:
d4: average improves from 2.5 -> 3.0 (+0.50)
d6: average improves from 3.5 -> 4.25 (+0.75)
d8: average improves from 4.5 -> 5.50 (+1.00)
2d4: average improves from 5.0 -> 5.62 (+0.62)
d10: average improves from 5.5 -> 6.75 (+1.25)
2d6: average improves from 7.0 -> 7.97 (+0.97)
The simplest fix, which underscores how crappy a feat it is, is that you do +1 damage once per turn. To make that actually worth taking, I'd change Savage Attacker to: "Your melee weapon attacks do +1 damage."
I have the same concern about Great Weapon Fighting. For simplicity and speed in play, a flat damage modifier is much better than adjusting some damage dice to treat a roll of 1 or 2 as 3. I'd keep it simple and give it +2 damage, which is consistent with the Dueling fighting style.