I searched around and couldn’t find anything better. Conceivably we could construct at transitive verb “lay” from the root KAY “lie”, but the usual transitive suffix ta already appears in the intransitive verb caita, so I am not sure how to construct it.

+Paul Strack yes, I was also thinking about constructions like *serta- "(make) rest".

Gnomish had pâ- v. “put; do put” but that hardly helps us.

It's a pity we don't know the meaning of http://eldamo.org/content/words/word-423129795.html ... In any case, I imagine that a transitive neo-Q derivation from KAY would probably need to involve an adjective (*caica/caiya/caira?) like tanca-ta- does.

*Sed-tâ- would yield Q *sesta- based on http://folk.uib.no/hnohf/qevolution.rtf .

A third potentially useful root is KHAD.

Actually, yā can also be used to form transitive verbs, so perhaps unglossed caia- could mean “lay” from KAY-yā > kaiya > kaia.

+Paul Strack Indeed what I was hinting at — but given the lack of a gloss, the odds (yā being intransitive in lauya- etc.) do not convince me to use caia- in that sense just yet.

Well -ta can be used both transitively and intransitively, so maybe -ya can as well. But you are right: without a gloss, it’s just a guess.