Is there a difference between the X5R camera?gimbal supplied with the Osmo Raw and the one supplied with the Inspire Raw?
If I recall correctly the X3 Osmo camera was not cmpatible with the aircraft while the I1 X3 camera was compatible with Osmo.
I'm wondering if it's also the case with the X5 series.

X5R is compatible both for Inspire and Osmo Raw. You can see a discussion in the Osmo forum.

DTK is right. You just need to flash the firmware of X5R.

You will need the X5 adapter for the Osmo (if you don't already have that) but otherwise it is the same unit used for both setups.