I honestly can't come up with an answer as to why someone from Earth and Mars would need to transact in real time. I suppose if you were sending remittances, having faster confirmation would be nice, but unless we develop FTL communication, it doesn't sound very feasible.

Even then, there's going to have to he relays out in space for when the sun is in between the planets.

I think you're right, local LN networks would be doing the bulk of the work.