I always travel with my Phantom in a Lowepro drone backpack and it works great. However, it’s usually always laying horizontally and not subjected to any vibrations to speak of. And I keep the gimbal in a cover/holder thing when not in use.

Next week, I’ll be going on about a 100 mile round trip motorcycle ride with a group of buddies and I’d love to bring my Phantom along.

This would mean carrying the bag as a backpack, putting the drone at a sideways angle and pretty violently bouncing up and down the whole time. I ride a stiff GSX-R too so that doesn’t help.

Will this be too much force and vibration exerted on the gimbal or will it be fine? I sure do baby the heck out of it and would hate to damage it in any way.

I have a hard shell backpack type case of a different make.   Looking at this one, the inner "packing" is very stiff foam.   As far as protection from lower frequency vibration (transportation), its pretty much useless.  I can see by the stiffness of the shell and the inner packing, that pretty much all of the vibration (and shock) would be transferred directly into the payload.   For my pack I would not do it.     Point being, if the inner packing was more forgiving, it  would be better at insulating it from vibration.  (dampening)   Stiffer materials are not good dampeners.   
One additional note,  many products are not resistant to vibration.  Some electronics are and you might see them designated as "ruggedized".   Typically for devices like computers, that can greatly increase the cost.  

Hello there slimjim837. Good day and thank you for sharing these information with us. I hope that our fellow DJI co pilots can give out some information based on there experience on how do they pack and transport there DJI Phantom 4 Pro. Again, thank you for your support and stay safe always.