Can a Unicorn have class levels?

[Fax Celestis]

Problem: ECL 8 as the base means that you're getting even less cleric out of a unicorn. 5 levels of cleric if you have it come in at party ECL, 3 if you have it come in at cohort level...

If you could convince your DM to swap its SLAs for natural cleric casting, and then let cleric stack the progression, it'd be worth it.