Highbeam already nailed it. No vinegar needed, but a vacuum is.
On the BK30's, is just about impossible to get it clean in place. At least in my stoves, you need to pull the combustor out to get a vacuum nozzle squarely onto it, and pull the ash out. The Princess may be different, but given that both likely have some debris hiding in the chamber behind the combustor, you'd probably do better pulling yours too.
I buy 10 or 20 feet of gasket at a time, as my combustor takes about 2.5 feet, and I pull it every summer (x2 stoves) for my yearly cleaning... sometimes once mid-season, too. If you already have gasket, slide that bad boy out, and give it a good vacuum. If you don't have gasket on-hand, I'd get some on order today, and just vacuum in-place as well as you can with an angled nozzle for today. Just be careful not to damage those relatively fragile steel cells.
Pulling and re-installing the cat is very simple, only takes a few minutes. Use a slotted screwdriver to pry on one of the corner ears (not the shim!) on each side, until you get get a finger behind it. Then grab and wiggle it out. The old gasket will be crumbly, usually 60% will come out with the combustor, and 40% will be left in the hole. Sweep it all up with your hand and throw it in the trash, that stuff won't burn.
Use your shop vac or ash vac to vacuum the combustor clean. Then take some masking tape and tape the end of your gasket roll to the combustor, starting at one corner. Wrap the gasket around the combustor, and trim flush with scissors. Then wrap the entire gasket in 2" wide masking tape, to hold it tight to the combustor, the tape all burns off in the first use. Slide it back home in the hole, replace the flame shield, done. Total job time is under 10 minutes, including fetching and putting away the shop vac.
The proper gasket is 2" x 1/16" interam. Used to run about $2 per foot, but latest price check had it up near $5/ft. I get mine from Woodman's Parts Plus.