The workaround is to have the actions you want shared tied to one of the hotbars that themselves can also be shared. You can actually set which hotbars are shared across all classes under Character Configuration > Hotbar Settings > Sharing. The default is that Hotbars 9 and 10 are shared, leaving 1-8 to be unique for each class.

I have sprint bound to the S key (I don't use WASD so this works for me.) Sprint lives permanently on Hotbar 10 in slot 6. Hotbar 10 is shared across all classes, and it's the one hotbar I keep visible at all times. Hotbar 10 slot 6 is what is technically bound to the S key, but because that hotbar never changes, it is effectively a permanent binding of sprint.

Other ones that live on the perma hotbar 10 include summoning and withdrawing the battle companion, limit break, materia extraction, mount roulette, and my Oh Snap Mechanics macro for alliance raid.