[RPG] Can you choose not to add your proficiency bonus to an attack roll

Once per turn when you roll damage for a melee weapon attack, you can reroll the weapon’s damage dice and use either total.

Roll all of the attack’s damage dice twice and add them together.

Emphasis mine. Savage Attacker causes you to reroll all of the weapon's damage dice, so if you used it with a greatsword that deals 2d6, you would reroll both dice and choose one of the two sets of results. In the same way, if something (such as a critical hit) causes you to roll the weapon's damage dice more than once, then you have the option of re-rolling all of those dice and choosing which to use.

The most optimal way to do this would be to play a Half-elf Hexblade.

Drow (PHB) & Shadar-Kai (UA Elf Subraces) come in second (+1 to Cha vs Half-Elf's +2), and other elves are still viable (just a little behind on Cha). You can start using a heavy weapon once you reach 3rd level in Warlock with the Hexblade Patron.

Hex Warrior (XGtE 55,56)

• Proficiency with martial weapons (as well as medium armor and shields)
• Select a single weapon which is not two-handed and use Charisma for attack and damage. This extends to your pact blade, regardless of its type.

Pact of the Blade (PHB 107)

• At 3rd level you can choose to summon any melee weapon as your Pact Blade. At this point you may choose your heavy weapon and attack with Charisma.

Also note that your attack cantrips such as Eldritch Blast and Shocking Grasp would benefit from the reroll as well.

Here are some ideas that come close but I don't think quite work:

• 3rd level Paladin with the Oath of Devotion
  • Sure, you add Charisma to your attack. That probably doesn't count as using Charisma. Let's call it a bonus. Note that you don't add Charisma to your damage. (Maybe you can convince your DM that this works. It comes down to what using Charisma actually means in the context of the various rules.)
  • Also, this takes an action where you could have been attacking and uses up your Channel Divinity that you only get once per rest.
• Kensei Monk with a Bow
  • Improvised weapons don't inherit properties from what they really are. Using a bow as a club/staff doesn't give it the heavy property and doesn't make it a monk weapon. In melee, a bow is just a length of wood (not even as good as a staff). (It certainly doesn't inherit the ranged property.)

Honestly, it makes sense that you are having a hard time using a heavy weapon without using Strength. This is part of the game balance that 5e sets up. If anything Hexblade/PoB breaks this balance by using Charisma with any weapon. Don't be surprised if you see Hexblade as the go-to for getting around balancing rules.