* shields are not just 'held' like a weapon; you cannot just let an equipped shield go to drop it, or equip it just by holding it. A shield only gives +2 to AC when properly equipped; just 'holding' one will not improve your AC. To either equip or un-equip a shield costs you your action.
So if your example PC were holding a weapon in each hand then he could easily drop one (not an action), grip the other in both hands (not an action), attack twice (attack action with Extra Attack), switch to holding his weapon in one hand (not an action), pick up the dropped weapon, or draw a new weapon (your one free object interaction this round), and can move as and when he like during all this (up to his move) and take any bonus action he qualifies to take.
The problem with the shield bash guy doing this is that it would take his action to un-equip the shield, but because he used his bonus action to shield bash and he only had that bonus action because he's taking the attack action, he cannot take any other action except the attack action, so the 'un-equip shield' action is not available this round. Even if it was, it would be pointless to get rid of your shield in order to use your weapon 2Hed because you wouldn't have an action left with which to attack because you'd've used to to un-equip your shield. Also, you'd need both your free object interaction to pick up your dropped shield AND your action to re-equip it.
