My group is just starting to dabble and question the mystic side of things. They have been fully enjoying the more scientific end of things so far
At it's core, the system seems to be, exorcism = Mystic Power test with the usual 0=possible failure, 1 = min success 3+ = major success. That should do in a pinch I suppose but it leaves a little something to be desired. As the exorcism is the player attempting to forcibly remove a spirit, I plan to make it a combat/opposed roll, mystic power vs mind points (or other relevant stats) and treating the results as such. When reduced to 0 mind points (or a preset value) the spirit is expelled and may be defeated properly or the story continues as desired. This would allow both the player and the spirit to use items/relics/artefacts/ect to boost their combat numbers. A holy symbol used to increase the mystic powers roll or the spirit using it's resources to distract and interfere with the dice roll.
A third option could also be that the player just needs to use a specific item in conjunction with Exorcism to automatically expel spirit (to skip the tedium of dice rolls if you are more into just the role play aspect). This could be viewed more movie style: 5th Element, Men in Black, Stargate, Guardians of the Galaxy.