Pics have me even more convinced the firebox is the problem.
Id cut a pie vent in the bottom of the box you have, then scab another section to the bottom to hold a ash pan, Then drop your grate down low. Or you could just cut a square hole out of the bottom and weld your grate right into that hole, then fab a 3 or 4 inch extention to add to the bottom with a front slide vent. . Either way, you wont have to make a new door .
Once again, you said blowing air through the vent got it going!!!