One of the key components that made Bring a Trailer so successful is bringing transparency to the industry. First to even be able to bid on the car you have to submit your credit card, and the second you place a bid it puts 5% of the bid on hold on your card. Transparency comes in as you can click any username and see what they have ever done on the history of the site. Bids, comments, selling, how long they've been a member, etc.
Due to this transparency shill bidding is going to be far less than any other site, such as Ebay for example. And most certainly lower than in-person auction houses. If shill bidding ever takes place on the site it is going to be from an individual selling one car and having his friend do it, not a reputable dealer. Because the after the individual sells the car he doesn't care what happens, he's done.
The dealer has to show back up, again and again and again. Bring a Trailer has actually been a good strainer, filtering the good from the bad. Bad dealers and wanna be dealers fizzle out, or are called out and questioned by the community. Even those that have registered under different usernames have been identified and had their auctions withdrawn by BaT.
I see your final question was how to deal with shill bidding, if you are selling a car, operate with integrity and don't ask anyone to place a bid as a catalyst for your auction. Represent the car honestly, accurately, and the best to your ability and let the chips fall where they may.