I've had a few inter-building connections have trouble with surges, most typically commercial PBX setups though (POTS lines, analog extensions, etc). Always ran network fiber between buildings, as Ethernet gear was always too expensive to 'roll the dice'. But these days there are more options for surge suppression for wired Ethernet. Though I'd imagine there'd still be the 300' limit that would make fiber necessary, even without surges. Kind of a shame there's no ethernet connectable Aux repeater.
You can, of course, use multiple main repeaters, but I'd worry about exceeding the limits of 100 devices per repeater; two total. Be sure to plan around the devices on the Aux being part of the count for their host Main repeater. I've run into a situation or two that had too many devices/rooms crowded onto the 1st repeater, making it impossible to add motion sensors/rollbacks without first moving stuff off to the 2nd main repeater. Leaving some 'headroom' during multi-repeater setups is a good plan.
I've owned enough boats over the years to know lightning is something to accept as LIKELY to happen eventually. Plan for it, either through prevention hardware or insurance, hopefully both.
But for this situation, yeah, I'd reach out to Lutron to see what their recommendations are for it.