After installing a mobile AC, I do have hot air intake from the front door
Those mobile A/Cs draw air from inside the room, cool it, and return most of it back into the room. Some air is used to cool the unit and is exhausted, in your case up your 'chimney'. This causes a small negative pressure, which will draw air from adjacent, unsealed areas (in your case, your front door).

There are dual-hose mobile A/Cs that use external air to cool the compressor, but that doesn't really help solve your current problem.

I would say that it's just to be expected; the price to pay for a cheap, mobile A/C.