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.)
- Improvised weapons does not grantYou 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-action attack from Two-Weapon Fighting, you need the feat that lets you dual wield-wield non-light weapons. (Based on the listed weight, we houseruledhouse-ruled it to be a heavy weapon.) Note that this feat also increases AC by +1.
- It is not a finesse weapon, so cantit can't be used for sneak attackSneak Attack.
- A dual-wield bonus action-action attack from the Two-Weapon Fighting rule does not add your positive strength(positive) Strength modifier to damage, unless you have the fighting style for this.
To sum it up:
- You MUSTmust have the Dual Wielder feat "dual wielder" for dual wielding-wielding non-light weapons.
- You SHOULD aqcuireshould acquire proficiency, either withusing the entertainer (gladiator) background or with the Tavern Brawler feat "tavern brawler".
- You SHOULD aqcuireshould acquire the fighting style "two weapon fighting" for strengthTwo-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 attack 1d4+str-action shield bash that does 1d4 + Str (bludgeoning) shield bashdamage on a hit.
- +3AC+3 AC. (2 from the shield, 1 from the dual wielderDual 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 shield-master to the mix for my next feat. Apart from the fact that I only have one bonus action per round, the benefits all stack.