Roosevelt settled for Republican surveying device (10)
I believe the answer is:
Here is my best explanation:
'surveying device' is the definition.
'roosevelt settled for republican' is the subsidiary indication.
I cannot really understand how this works, but
'settled' could be 'lit' (I've seen this in other clues) and 'lit' is present in the answer.
This may be the basis of clue (or it may be nonsense).
(Other definitions for theodolite that I've seen before include "Instrument", "transit, in America", "What a surveyor might use", "surveying instrument", "Surveying instrument" and "Surveyor's instrument".)