Venting Question for Masonry Heater Owners & Fireplace Gurus

Have to say that the masonry heater that we had constructed in our home is one of the best investments we could have made! Love-love-love it! Heater works and heats as it should.....as long as you never open the doors after the fire is lit. That being said, I have a question for those of you much more knowledgeable than I/we. We have a problem with smoke and ash billowing into the house when the door is opened to add wood. In investigating this, we have come to the tentative conclusion that the makeup air piping of 4" in diameter (which runs under the concrete slab and radiant heat tubing) is not large enough. According to the geothermal heating guys (who we talked to when it was suggested that there might be a negative pressure issue - house is definitely NOT sealed tight yet), a 4" pipe is only capable of moving something like 30cfm of air and what should have probably been installed was more like an 8" pipe which I understand can move an exponentially greater amount of air and would correct this problem. At this point, digging up the concrete slab and severing heat lines is not sounding practical, feasible, or appealing. Do any of you happen to know of some sort of fan or other solution that could be used to boost the airflow to send the smoke up and out?? We have a four socket electrical plug on one side of the fireplace that we could plug into if needed.
Have included some construction pics for reference. Cleanout above mantle and below heated bench on front, electrical on side without heated bench, two cleanouts on back. Top ring has since been attached to ceiling and doors are on.

Hope that all makes sense! Sincere thanks to you all in advance for any help that you are able to offer.

Thank you so much for your thoughtful response, but no, the top flue is not the issue. It is plenty big (8 or 10" I think?) for what should be going out. Also, firebox measures approx 20" deep and about 34" across the doors. We generally load it with roughly 50 lbs of wood, twice a day. Plenty of heat. The issue we think that we are having is with the make up air coming in to the firebox from outside of the house. The smoke fills the firebox completely when you are trying to start the fire, forcing us to resort to firestarters (like a tiny Duraflame thing) to start the flame. If you attempt to start a fire with paper and it doesn't catch the first time, then you have to either wait until ALL of the smoke dissipates or suffer the consequences of smoke billowing into the house. Same goes for if a lit log rolls over and you wish to shove it back into the pile before going to bed - smoke and ash in the house.

So, here is the theory that we have. Inside the firebox there are two tiles, one on either side, with a hole in them. These are tied into a 4" pipe, which, as previously described, runs under the slab to a vent outside of the house. This way makeup air is brought into the firebox and allows the fire to have enough oxygen to burn well when the doors are closed. We suspect that the pipe is undersized for the amount of makeup air that the fire needs to draw, thus not allowing enough air movement to allow the fire to expell as efficiently as it should, and making it tough to start the fire properly (without it choking itself out). Does this make better sense??

So, our question again is.....is there something that we can do to aid/boost the air movement and help to make this unit more efficient without having to jackhammer up the slab, cut heating pipes and fork out alot of money?? Thanks again for your input and help!