There are exactly zero options for training and taking the check ride in an LSA anywhere around where I live. I have a PP ticket, but am now flying with Sport Pilot privileges. The best advice I can offer you is to find one, two or three like minded individuals and buy an airplane. You can buy a nice E-LSA RV-12 now for under $70K, which works out pretty well split between two to four people. Operating costs are very low -- my co-owner and I figure $30 per hour wet, including maintenance and a reserve for overhaul (which realistically we'll probably never do). Form an LLC to own the airplane, everyone owns part of the LLC. That way it's easy to add or subtract owners. Yes, it will involve some initial outlay of cash, but when you figure the huge difference in hourly cost you'll recoup that - not to mention you can sell the airplane or your share in it.
If you don't want or need to travel cross country, there are planes that are MUCH cheaper to buy, but more costly to own and operate. Champs, Chiefs, Ercoups and so on. Might not be the best choice if you're old and chubby like me, but they're out there.
Light Sport definitely needs more exposure and better PR, but I wouldn't blame EAA for that.