Curious:
I'm glad you have made a break-through the carbonate solids build-up in the shell side of your BEM unit. Acid is capable of removing the solid mass - albeit, very slowly.
My experience in doing an acid wash is: Be VERY, VERY careful in pumping in acid solution into the shell side. You are generating CO2 gas as you dissolve the mass of carbonate solid inside and the pressure can build up quickly if your outlet is partially plugged up with solid particles that break out from the main mass. The result can be a massive gas eruption out of the shell side outlet and if you are not careful and have everyone strictly dressed in proper acid-proof clothing and shields, you could have a serious acid-related incident.
Don't push or try to accelerate the break-up of the carbonate mass inside. Let it take its time and dissolve and breakup slowly so the pump doesn't build up any back pressure. Use only HARD piping on the exchanger outlet, leading to a dedicated sump or vessel where you capture the acid solution for recyle and separation of the dissolved gas. Don't forget: you are creating a 2-phase flow - or worse, a 3-phase flow.
Good Luck.