If you wax โ ZFC-recommended path. 138 km sits comfortably inside any top wax's single-application life โ ride it out on one coat. No mid-ride lube, no oil in the jersey. If an afternoon Dolomite thunderstorm rolls over the Sella, the wax treatment rides through it: over a 138 km single day the cost of rinsed wax is small next to the post-ride strip commitment of oil-over-wax. Wear + efficiency are the priorities; faff is mostly irrelevant for a single-day.
If you're on oil โ pragmatic equivalent. One night-before application of your usual oil is plenty โ 138 km is well inside any decent oil's single-application life. No top-up bottle needed. Even an unexpected shower is a non-event since you'd already plan to clean and re-oil at home.