I don't see anything that prevents them from taking the feat. So far as I can tell a cleric could also take the feat Varisian Tattoo which would net them a +2 CL for all spells of the school chosen with Spell Focus.
A cleric however, would not be able to take the Bloatmage Prestige Class since that requires being able to cast arcane spells.
RAW allows divine casters to take Bloatmage Initiate, since you only need one of the Spell Focus feats to qualify.
You and your table should consider the divine ethical questions raised by doing this, particularly in regard to self-mutilation. Some of this refers to drawbacks of the actual Bloatmage prestige class; even though divine casters can't enter the PrC on their own, you're still borrowing from a path whose adherents have those abilities and drawbacks.
- Self-mutilation tends to be frowned upon by most deities. Only Zon-Kuthon and Lamashtu really embrace the practice.
- Even though bloatmages have a philosophy of "self-perfection," Irori would still reject the so-called discipline because of its risk of uncontrolled berserker rage.
- As long as it's magic, Nethys doesn't really seem to care.
- Gorum would tend to not support them, because he's all about physical combat.
Etc.
I think Asmodeus would be fine with it, he would just prefer you came to him for a contract for some extra power, instead of taking it into your own hand.