Who would you pay for sending messages? That's your centralization point. Alternatively if you allow "starting balance", how would you prevent from making a lot of accounts for spam sending?
Ontop of that, I think payment isn't critical. You join the mesh because you want to use it yourself - all you need then is to limit how much power you're prepared to spend on it. What does it matter to you if 100 people use your phone or none? ....other than power.
To put it another way, I think money would introduce a commercial motive which would end up gobbling up the system like bitcoin mining.
I don't think relaying messages would require that much power and as you said, "You join the mesh because you want to use it yourself".
"Just encrypt things" might be your reply. TOR folks have been fighting an uphill battle for ages with that as their main weapon.
LN payments are based on Sphinx[1], but they don't leverage the full capabilities of it. Sphinx only allows for single-use replies.
The proposals for bidirectional anonymity involve using HORNET[2], which builds on Sphinx (specifically, it uses Sphinx's single-reply message to establish a HORNET connection), and enables sender-receiver anonymity and bidirectional transfer beyond single-reply messages.
It has also been proposed to include the use of TARANET[3] to prevent deanymization via traffic analysis.
[1]:https://cypherpunks.ca/~iang/pubs/Sphinx_Oakland09.pdf