In RAW there appears to be no reason why a conjured spirit with the Innate Spell power couldn't be given a spell the conjurer knows as fetish-limited. Would it need a fetish to cast the spell? Or just not be allowed?

Don't see why it has to be limited the same way as the conjurer. This is a spirit power, after all. It doesn't work exactly the same as magic does for metahumans. You're not giving it your powers, it's using its own version of the spell.

It if it's using its own version of the spell, why can it only use versions that the conjurer himself knows?

The real question, however, is whether your spirits of man can cast spells with your Mentor Spirit bonuses.