LN doesn't record every TX on the base layer. Just the opening and closing of a channel.

Of course, I don't understand your point though. I'm not suggesting the lightning payment directly pay fees for that timestamp. Peter would accumulate lightning sats received in this way and periodically loop-out into his opentimestamps wallet, which would then help pay for opentimestamps on-chain transactions in the future.

Yes, that's exactly what I do with donations. Accepting LN for speeding up transactions makes sense and I'll probably do it at some point. It does require a bunch of code to do well though.