Roosevelt settled for Republican surveying device (10)
I believe the answer is:
theodolite
'surveying device' is the definition.
'roosevelt settled for republican' is the wordplay.
I cannot really understand how this works, but
'settled' could be 'lit' (I've seen this before) and 'lit' is located in the answer.
This explanation may well be incorrect...
(Other definitions for theodolite that I've seen before include "transit, in America" , "Surveying instrument" , "What a surveyor might use" , "Surveyor's instrument" .)