This word analysis focuses on transitive verbs meaning “put, place, lay, set”. Finding a good choice for these words is a challenge, and requires some invention.

We have some good options for these senses from the earliest versions of Tolkien’s language:

1) transitive ᴱQ. kaita- “to place, lie down“ vs. intransitive kaya- or kay- “to lie” (QL/46, PE16/75), both from the root KAYA “lie, rest; dwell”.

2) ᴱQ. kasóro- “to make sit, set” from the root ÐORO “sit” (QL/43).

3) G. pâ- “to put” and panta- “to arrange, order, settle; to set, put, place; to write a book” (GL/63), both likely from PANA “arrange”.

Of these, the roots KAY and PAN survive into Tolkien’s later writing. From the Etymologies we also have:

4) [rejected] ᴹQ. esta- “to place, set, plant” from [rejected] root ES², appearing beside a [rejected] noun esse “place” (EtyAC/ES).

5) ᴹQ. panya- and N. penia- “to fix, set” (Ety/PAN).

We don’t have any good verbs in this semantic space in Tolkien’s later writing, however.

The most recent unrejected word with an appropriate seems to be ᴹQ. panya- and N. penia- “to fix, set”. The root PAN retained the sense “arrange” in Tolkien’s later writings, and the glosses “fix” seems to imply this verb means fixing in place rather than simply putting or placing an object. Similarly, salvaging G. pâ- and panta- is difficult. G. panta- would need to be reformed as N./S. panna- which already has several other attested and important meanings: “to open, enlarge” (Ety/PAT) and “to fill” (Ety/KWAT). A verb form like Q./S. pan- is very close to the root and probably should mean “to arrange”.

Q. caita- survives, but Tolkien is very emphatic in his later writing that this verb is intransitive: meaning “to lie (down)” rather than “to lay”.

We do, however, have an alternate, unglossed verb caia- (PE22/159). This verb is marked OQ., but it could have survived in Quenya, could conceivably have a transitive sense “to lay”. This would be a reversal of the transitive/intransitive senses from ᴱQ.

The noun esse “place” was rejected and seems to have been replaced by Q. nómë “place” (VT42/17, WJ/206), perhaps from an [unattested] root NOM. This could have a transitive verb form nom- or nonta- “to place”.

Finally, the root ÐORO “sit” seems to be replaced but KHAD “sit”, which might also have a transitive verb form hasta- “to make sit, set”. This conflicts with (unattested) Q. hasta- “to mar” deduced from adjectives alahasta “unmarred” and hastaina “marred” (MR/254), but that verb is rather obscure making a homonym more tolerable.

On the Sindarin side, we might cognates nov- or nonta- “to place” and/or hasta- “to make sit, set”. A cognate of caia- might be caea-, though figuring out how exactly to inflect the Quenya or Sindarin verb forms would be a challenge.

Any of these are candidates for neologisms for transitive verbs meaning “to put”. At the moment, I am inclined to keep them all of them with slightly different meanings:

1) ᴺQ. nom, ᴺS. nov “to place, put” from the (unattested) root NOM “place”, with an emphasis of putting into a specific location or place.

2) ᴺQ., ᴺS. hasta “to set, (lit.) to make sit” from the root KHAD “sit”, with an emphasis on the action of placement.

3) ᴺQ. caia-, ᴺS. caea- “to lay (down)” from the root KAY “lie”, with an emphasis on the downward motion and placement in a horizontal position.

EDIT: Based on conversations with Shihali (on the Discord server's +#eldarin channel) and Tamas (see below) I've made the following changes:

1) ᴺQ. satta-, ᴺS. satha- “to place, put” from SAT “space, place”, with an emphasis of putting into a specific location or place.

2) ᴺQ., ᴺS. sesta- “to set, (lit.) to make rest (on)” from the root SED “rest”, with an emphasis on the action of placement.

3) I am leaving alone for now.

#1 was changed to derive from an attested root, #2 was changed to avoid conflict with hasta- “to mar”.