I would deepen the pan to fix that problem. If the bottom of the pan was about 1/16" to 3/32" lower than the vent hole it will improve the situation a great deal.
Now, by going with a vent liner I have no doubt that the speed the gun would fire at would be much better than it is but that's going to cost some money.
As for needing a gunsmith that specializes in black powder guns, that would be good if one can be found. On the other hand, any gunsmith worth his salt should easily be able to install a liner at the right location. The problem with that is, many gunsmiths that work only on modern guns don't know where the right location is.
With this in mind, if I was going to take it to a modern gunsmith I would first buy the liner. Although a White Lightning liner is probably the best, it takes a special tap with a 32 thread pitch. I've used dozens of the liners sold by Track of the Wolf with standard threads on them and they also have the short vent length of the White Lightening liners and they work fine.
After getting the new liner I would then, make a mark on the side of the barrel that is in line with the top surfaces of the pan. Then, I would make another line that is perpendicular to the first line that is centered in the middle of the pan using a pointed punch or even a good sharp nail.
Just to make sure there was no mixup in the location of the new vent liner I would use my nail or punch to make a indentation where the two lines crossed. A good rap with a hammer on the nail should make a noticeable indentation.
Then with gun in hand I would go to the local gunsmith and ask him if he could install the liner exactly at the location of the indentation.
He will need to have a good milling machine to do this job to have the ridgidity needed to keep the tap drill from walking off of the exact location because the drill bit will be making an interrupted cut due to the existing hole.
Of course, that is what I would do. Not what someone else would do.