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 grant proficiency bonus, unless you are proficient with using them (as weapons). (So ordinary shield proficiency is probably not enough. I get proficiency through the entertainer gladiator background.)
- It is not considered to be light, so for bonus action attack you need the feat that lets you dual wield non-light weapons. (Based on the listed weight, we houseruled it to be a heavy weapon.) Note that this feat also increases AC by +1.
- It is not a finesse weapon, so cant be used for sneak attack.
- A dual-wield bonus action attack 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 feat "dual wielder" for dual wielding non-light weapons.
- You SHOULD aqcuire proficiency, either with entertainer (gladiator) background or with the feat "tavern brawler".
- You SHOULD aqcuire the fighting style "two weapon fighting" for strength modifier. (One level of fighter is how I do it.)
If you do all this, then you get:
- One bonus attack 1d4+str (bludgeoning) shield bash.
- +3AC. (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.