You can always pull the stove to install a pipe damper but maybe you can get it in with the stove in place..hard to tell from here. Can you reach your hand up into the flue opening? If so, and you can get a small drill between the top of the stove and the fireplace, drill a hole with a long bit (holes a little larger diameter than the damper shaft of course) through the front and back of the liner just above the flue opening. The holes will be toward the front and back of the stove so that the damper shaft is running front to back in the fireplace. Then you will need two people (and quite a bit of time to fiddle with it and cuss at it,) one holding the damper plate inside the flue and the other inserting the shaft through the hole in the front, the damper plate guides, and finally the rear hole. Depending how much larger the holes in the pipe are than the shaft, it may require tilting the shaft up or down, so you need room to do that. You can do a trial run with the shaft only, no plate, to determine that. I have only drilled connector pipe, not a liner..could be harder to do.
In addition to big splits, you might try shoving the coals to the back, then the fire would have to move from back to front against the air flow..maybe that would slow it down some, I don't know.
In addition to big splits, you might try shoving the coals to the back, then the fire would have to move from back to front against the air flow..maybe that would slow it down some, I don't know.