No, if you try to Bike when she just wants to regrab, you'll keep your Bike. The reason Bike works is because dsmash' hitbox is frame 21, though ZSS really only has a 20 frame advantage. However, NOTHING in your arsenal really activates on frame 1 except for jump (which, in 1 frame time, doesn't net you the distance to get out of dsmash range) and Bike.
The Bike gets into the game starting at frame 1, but won't become a hitbox until frame 16. Once it gets out though (frame 1) it kinda distorts Wario's hurtbox and generally messes with a lot of physics the game has to offer. The Bike doesn't feel pity or remorse, and will cancel out all sorts of stupid attacks and projectiles and it cannot be stopped unless killed Wario jumps off it. (edit: or Wario get knocked off it, of course...)
It can really be used in really weird situations, I've heard (though never tried cause it warrants more failure than it ought to) it can be used to get out of Snake's nair (which probably works, Marth can upB out of it as well which means the individual hits don't really combo into each other), you can pull out your Bike while being chaingrabbed by Falco (though that's never really a good thing), you can use it to cancel out those annoying aura spheres Lucario sends after you trying to catch your landing frames (really, you can do that **** on reaction).
Anyway, Bike works against the dsmash. But if she can also do a normal regrab (as noted) which is rather gay and annoying and generally a huge waste of time cause it's rather difficult to do and eventually you'll just take 15% worth of pummels and then you get a free fsmash cause they **** up. If they don't **** up, then it's all the more gay cause they'll be grab releasing you for 3 minutes straight until you finally get into kill percentage.
Oh another thing against ZSS concerning the Bike: if she starts camping with neutral B, you can just Bike right through it. If timed well, you'll catch her in her cooldown lag (meaning: guaranteed 9% **** yeah (no really)).
loool wall of text