Is there a way to log contacts over a mesh network, using TQSL/ LOTW/ EQSL/ etc? If so, how should it be logged.
I'm not sure you can or should log contacts that were not directly between your stations.
Contacts through repeaters, packet reflectors, mesh networks, IRLP, echolink etc. don't really count as a measure of your skill and station. Satellites are an exception though.
QSLs are generally not exchanged for contacts made through repeaters (other than orbiting satellites), nor for those made through such media as Echolink. Such contacts, fun though they are, generally are not valid for operating award purposes.
As LotW is a QSL system for confirming awards and achievements, it's not really appropriate to log contacts made via repeater or mesh.