Very Senior Member
Given that the OP is running power on the same route I would install a small water proof enclosure mid span and put an inexpensive switch ($15 ) in the enclosure to act as an amplifier/repeater. Put some surge protectors at both ends of the run for a few bucks more.
The power to lights will be ON/OFF, not a constant power supply. The idea is not bad, but needs additional constant power line. One outdoor application switch on the furthest pole, not in a box underground, may solve the CAT6 cable problem. PoE powered outdoor switches are available too, but they need PoE in the house. A switch somewhere in the middle may allow future network expansion - IP cameras, outdoor AP, branching out to another area, etc.