Hello lockeal, and welcome to the forum.
The bottom line is that coal needs a lot of air, many times more than wood from my experience.
I went from a 6" chimney last year to an 8" chimney this year, and I added about 5-6' in height. The difference has been that the boiler's combustion blower rarely comes on once the fire is stable. The added draft from the bigger and better chimney is doing the job.
If you are going to do anything to your stove, I'd add a combustion blower. They are less than $100, and really help getting a fire going and if hooked up with either an aquastat [for a boiler] or thermostat for a stove or furnace, the blower can stabilize heat output as well.
I'm sure your chimney will draw much better once the weather cools down. Cooler weather is just a few weeks away!! There is a simple check for draft in your chimney and stove: when you open the loading door, with either a wood or coal fire, do you get a roomfull of smoke and fumes?? or is there just a little smoke getting out then it is pulled up the chimney?? With my earlier too-small chimney I had smoke getting out into the room in huge quantities, now I get only a little smoke when I first open the door.
You said in your post that you had tried a lot of things with your attempts to get coal to burn. I too have struggled at times to keep a fire going or not burning out.
The main thing with coal is air flow, and heat from the coal bed. You need to have as deep a fire [pile of coal] as possible. I can get about a 12 hour burn in my firebox if I heap the coal on to a depth of at least 14-16". My fire burns about an inch an hour. If you don't have more than say 6" of coal on the fire, once it burns down to a thin or shallow fire, it starts needing more fuel, the heat in the pile will drop and the fire will go out. Leaving some unburnt coal behind. The fire needs to feed on itself. A single piece of coal usually won't burn, it needs to have heat from other coal next to it.
It is rare in my boiler to have a fire burn all the way down to just ash. If I'm letting the fire burn out, there is always some coal that would have burnt longer and more completely in a coal bed that is left behind.
Your grates need to be able to let the burnt ash fall through into the ash pan as the fire is burning. Otherwise the ash will clog the grate, block the air flow and smother your fire. If you go out and shake the grates at 3-4 hour intervals, does the fire keep going? Are you trying to get a load of coal to burn all the way to ashes like you would a wood fire??
I hope some of the above is helpfull, I wouldn't reline the chimney unless nothing else works.