Be careful how you use the LSA description. The DA-20 DOES NOT fit that designation despite being lighter than the TB-9s that Parks replaced. I've flown both and the eclipse is a better aircraft in many respects. It's much more efficient and handles much better. In four years of flying them as a student and instructor, I had more than two people in that plane exactly three times. The DA-20 also has a 10 year track record behind it already so it's hardly a new design. However unlike a tampico, it's purpose built. The TB-9 a TB-20 with a shorter wing and about 100 fewer ponies. I've also seen DA-20s take abuse that would put a tampico out of commission.
As far as doing your private in an aircraft that's approved for LSA, why not? Tens of thousands have received their private in J-3 cubs, luscombes and other aircraft that fit the designation.