One way to tie it to the lottery and still have all tickets equally likely to win is by giving ticket numbers sequentially and mapping the tickets onto the possible national lottery outcomes. This is easiest if the total number of tickets is fixed in a way which makes it easy for people to know if they've won or not.
For instance, if the number of tickets are limited to 59 (numbered 1 to 59), then the winner is simply the person whose ticket number is the same as the first ball that is selected on the main lotto draw, and subsequent balls are ignored. If that is too many tickets, then limit the number of tickets to 39, and use the first ball in the thunderball draw.
If you want to issue more tickets, then there could be 78 tickets (=39*2), which are numbered 1a, 1b, 2a, 2b, 3a, 3b ... 38a, 38b, 39a, 39b. Then, the number must match the first ball in the thunderball draw, and of the two people with tickets that match this number, the winner is decided by the second ball drawn; 'a' wins if the second ball is in the top half of the remaining balls, 'b' wins if the number on the second ball is in the second half of the list of remaining balls.
I think this is a pretty simple system which uses the lottery and can't be rigged.