You can remove the handbrake leather grip by loosening the 2x allen grub screws on the underside. That'll expose the button, but you still won't be able to see the rod/mechanism that releases the ratcheting pawl, since it's between the 2 halves, and they're riveted.
If that mechanism isn't working, you likely will just need to replace the whole lever (part A132J0147F).
And as you surmised, if you have to increase effort or clicks to apply the e-brake, chances are you either need to adjust or replace the parking brake shoes, or one of the many cables in the parking brake system is binding.
Be careful applying too much pressure to the handbrake lever; the OEM leather boot is rather small and tight-fitting, and you can easily snap the plastic gaiter (A132V0071F) with too hard of a yank.
2011 Evora NA - Phantom Black | Paprika | MT | 2+.5