I agree with Mike. However, if you got a regional job offer, then other regionals will be calling you soon, it always seems to work that way.
If this is not your dream airline job, then try to delay the class date. It is always adviseable to leave an employer on good terms if possible.
If you have to take the class date, then try to negotiate with the boss...a few thousand dollars is a small price to pay for not having a black mark in your employment history. Few potential employers would be very concerned with your leaving for a better job, but if the boss is vindictive (sounds like he might be) then he might poison future reference calls by commenting on things unrelated to your real reason for leaving...such as your poor flying skills and substance abuse habits....
I think that there isn't much chance of him suing you for business losses unless you are an independent contractor and your contract stated that you would work for a period of time and perform certain defined tasks. Are you an employee or a contractor? I'm pretty sure you're an employee. Most training contracts don't require you to stay or do anything in particular, they just say that if you choose to leave early you will repay some training costs.
Also, training contracts are usually considered to be coercive if you were offered a job, quit your old job, and then were forced to sign a contract as a condition of hire. Did they provide a copy of the training contract to you before you quit your old job?
If you have to go to battle with this guy, you should talk to an employment attorney to determine your exact contractual situation. Many shady employers (including certain airlines) have you sign contracts that they know are not enforceable in court just to have ammo to intimidate you.