Only via Grin and on-chain transactions for now.

Using the Lightning Network just to make one payment to purchase the copper membership isn't worth it. Unless you are using a custodial service like Blue Wallet that doesn't require the user to open his own channels. But even then you need a route to the copper membership node and Blue Wallet doesn't have one.

If you are opening and funding your own channel, you will have to pay a standard fee to open it. After that, you pay a base fee and fee rate (standard routing fee + a percentage of the total value being routed). To finalize the transaction, you need to broadcast another standard on-chain closing transaction.

To sum it up: You have paid 2 on-chain transactions instead of 1 and you paid additional LN fees on top of that.  

I suspect the time involved would not be justified by the number of people who would use it. Likely implementing Lightning would mean more users asking for help, and more things that theymos would have to solve manually. Neither of which I would guess are high on his wish list.