What happens if you manifest a Zoetic Cavern?
Yes, you can use the morph cost of the Cavern.

In the rules for manifest rule 701.33c says:


  If a card with morph is manifested, its controller may turn that card face up using either the procedure described in rule 702.36e to turn a face-down permanent with morph face up or the procedure described above to turn a manifested permanent face up.


So if you are turning a card face up using the morph procedure, the manifest procedure and its restrictions become irrelevant.

Manifest is a separate, though similar, mechanic from morph and it is incorrect to call the cost of turning a card face up using the manifest procedure a morph cost.