This root was the basis for horse words starting with the Qenya Lexicon of the 1910s, where it appeared unglossed as ᴱ√LOPO (QL/56). It was compared to the root ᴱ√LOQO, and these roots include the verbs ᴱQ. lopo- “gallop, run (of animals)” and ᴱQ. loqo- “run (of human beings)”, so I think it is likely those were the meanings of the roots as well (QL/56). In the Qenya Lexicon, ᴱ√LOPO had the derivatives ᴱQ. lōpa “horse or mare” and ᴱQ. lopsi “mare” (QL/56), but elsewhere in that document there was the word ᴱQ. lapatte “rabbit” < ᴱ✶lopatte, so I think that word is related as well, though Tolkien did mark the primitive form with a “?” (QL/51). The probably-related word ᴹQ. lopo “rabbit” appeared in a discussion of nouns from the early 1930s (PE21/31).
In The Etymologies of the 1930s the root ᴹ√LOP was glossed “horse”, and its derivatives ᴹQ. olombo and N. lobor had only that sense (EtyAC/LOP), but horse words were usually derived from ᴹ√ROK from the 1930s and later. The word Q. lopoldi “rabbits” appeared in some 1965 notes on Númenor, so It seems likely √LOP remained valid in some form. Another possibly related late word is S. ✱laba- “hop” as in S. Labadal “Hopafoot” (UT/60).
Neo-Eldarin: For purposes of Neo-Eldarin, I think it is worth retaining ᴹ√LOP with a verbal sense “✱gallop, lope”. In this way, it can remain the basis for rabbit words based on their loping run. The horse words from this root might be specialized for heavier loping horses like destriers, since ᴹQ. rokko was said to refer more specifically to a “swift horse” (EtyAC/ROK).