i personally don't look into this research often, but i did have a terrible xp with post LP headache which i would not recommend to anyone needing to fly. i had about a week from dx to a round the world kind of flight and i don't know what i would have done if that LP headache was still in the mix at the time.
i am not sure whether researchers have gotten around to concurrently examining multiple factors which may come into play, such as needle gauge, bevel orientation, associated nature of puncture inflicted, patient nutritional status (where wound healing is concerned in particular) and length of time spent prone post procedure.
for my 2c, would be a good idea to
1. read up on needle gauge, tip configuration, needle bevel orientation in relation to dural fibres, and so on. couldn't hurt to read this
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2660496/ plus any more recent updates (i don't agree with everything in that article as you'll see below)
2. before your procedure, make sure you are topped up on nutrients that support wound healing, clotting and so on. esp where there's overlap with nutrients known to be low or imbalanced in MS. anything you can do to minimize chances of a slow to heal CSF leak.
3. take as long as they suggest to stay prone after procedure at the hospital, and then make your own decisions re further self prescribed bed rest duration once you're home again. (repeated CSF pressure changes from being intermittently upright and prone doesn't sound like a recipe for a smooth healing process).
personally, i lay down for as long as they said to at the hospital, went home and then over the next week, staggered around with a crippling headache (sitting, standing, anything where spine was vertical or close to it, it hurt)
by the end of the week my neuro said '
rest this weekend and if it's not gone by monday, blood patch.' i did NOT want whatever this blood patch thing was. so i laid flat on the couch pretty much around the clock for the weekend. it did the trick. no blood patch required.
i don't know why they say lying down after puncture but before the headache occurs is of no consequence (study design problem?), only to prescribe bed rest anyway once the headache has already occurred. at least you're out of their bed by then??

ha
if you're going to end up with this headache no matter what you do, then you don't want to be stuck upright on a plane for the worst of it. vesta's suggestion re rescheduling is good - it's not really worth the gamble. can't hurt to do all you can to prepare once you do eventually have an LP done. i personally wouldn't say no one should have an LP because my xp was bad. at the time i was deficient in nutrients required for wound healing. i just didn't know it. if i had, maybe there would not have been such an issue.