How can I get a neutral wire for this switch?
Somebody really made a hash of things here

Whoever put all this in really made a mess of things.  That low-voltage box they used for the switch was utterly inappropriate for the task; furthermore, they ran wires through the hole in the lower box to wire up the upper switch without a conduit to protect them, drilling a hole in the lower box in the process I am sure.  Also, they managed to pigtail the GFCI's earth grounding wire to the incoming hot(!), which explains why its mounting screws were electrified.  Finally, the presence of the GFCI means that while everything is on one circuit, you need to make sure that the smart-switch hot and neutral wind up on the same side (line vs. load) of the GFCI, lest you inadvertently turn your smart-switch into a remote-controlled GFCI tester!

I would replace this all with a single two-gang box, mounted at the current GFCI's location, with the GFCI and switch sharing it.  The GFCI LINE HOT goes to the junction of the black wires from the upper left and middle right cables in your diagram, while the GFCI LINE WHITE goes to the junction of the white wires from both those cables.  Then, GFCI LOAD HOT goes to the black wire from the lower left cable in your diagram, while GFCI LOAD WHITE goes to the junction of the white wires from both the lower left and upper right cables in the diagram, and the GFCI's green ground pigtail gets connected to the junction of the bare wires. You'll want to get new wirenuts as well, since at least one of the junctions in the GFCI box looks to have been made in a completely improper fashion.

Now that that's straightened out, we can add the smart-switch.  The smart-switch line wire gets nutted into the GFCI LOAD HOT junction (putting the outdoor lights on the GFCI's load side as they were previously), while the smart-switch load wire gets nutted to the remaining black wire, coming from the upper right cable in your diagram.  The smart-switch neutral, then, gets nutted to the GFCI LOAD WHITE since your lights are on the load side of the GFCI, and last but not least, ground on the smart-switch gets tied into the existing bundle of (bare and green) grounds.

Once all that's done, patch the drywall as needed, button the new box up with an appropriate two-gang faceplate, turn the breaker back on, and enjoy your new smart-switch!