My character uses this technique, and I believe we are doing it both RAW and RAI. So here we go:
- Any object can be used as an "improvised weapon".
- Because a shield is not obviously similar to any listed weapon, the damage is 1d4.
- The shield does not cease to grant its AC bonus when used as a weapon. (I think this balances out the weaknesses quite well, and I would not do this otherwise.)
- You don't add your proficiency bonus to the attack roll for improvised weapons, unless you are proficient with using them (as weapons). (So ordinary shield proficiency is probably not enough. I get proficiency through the entertainer variant of the gladiator background.)
- It is not considered to be light, so for the bonus-action attack from Two-Weapon Fighting, you need the feat that lets you dual-wield non-light weapons. (Based on the listed weight, we house-ruled it to be a heavy weapon.) Note that this feat also increases AC by +1.
- It is not a finesse weapon, so it can't be used for Sneak Attack.
- A bonus-action attack from the Two-Weapon Fighting rule does not add your (positive) Strength modifier to damage, unless you have the fighting style for this.
To sum it up:
- You must have the Dual Wielder feat for dual-wielding non-light weapons.
- You should acquire proficiency, either using the entertainer (gladiator) background or with the Tavern Brawler feat.
- You should acquire the fighting style Two-Weapon Fighting in order to add your Strength modifier to the damage. (One level of fighter is how I do it.)
If you do all this, then you get:
- One bonus-action shield bash that does 1d4 + Str (bludgeoning) damage on a hit.
- +3 AC. (2 from the shield, 1 from the Dual Wielder feat.)
Considering the investment (one feat, one background feature, one fighting style), I think it is well balanced.
I might add the Shield Master feat to the mix for my next feat. Apart from the fact that I only have one bonus action per round, the benefits all stack.
