I experienced the same problem last winter when spooning a set of michilen road pilots on my Triumph. The back tire went on perfect but , the front was out of round. I sent the tire back and they sent me a new one. I put it on and had the same problem. I talked to a local bike shop owner and he said to spray silicone around the entire bead and inflate it to 90psi. I am in no way saying that a moped tire is meant to hold this kind of pressure...It may blow up and kill you, blow your eardrums out , what have you.
Another thing to try is to inflate your tire to max pressure and bounce it around the entire tire a few times. Deflate the tire to just enough pressure to keep it up and bounce it again, repeat this cycle a few times. I would try this first.
Dumb question but, does your tire have bead locks?
There is a small molded line on the tire that runs just outside the circumference of the bead. Is this line the same distance away from your rim the entire way around the tire. If it is, you have a tire problem. If it is even a millimeter or two off, you have a seating problem.
I guess you cleaned your rim properly when you took the old tire off?
Are there any scuffs on the inside of the bead seating surface on the rim? You used rim protectors right??? Don't worry if you made this mistake...I have. Sand down any scuffs on the rim. They can keep the tire from seating the way it should.
Thats about all I can think of right now. Hope this helped.