You will need to tank 200 full party duties as a DRK for the War Panther, but after that, you can ride it as MNK, as any class.

And you don't really need to worry about tanking for real. If you want the mount without actually having to tank anything, you can join any of the many "tank mount farm" parties in the cross-world party finder and just do something like... Garuda Ex 200 times... It will be very boring, but especially with new Stormblood gear coming and making people ridiculously strong, there's zero chance you'd have to actually tank anything.