Insist, and offer to keep deadlock (5-3)
I believe the answer is:
stand-off
'deadlock' is the definition.
(I've seen this before)
'insist and offer to keep' is the subsidiary indication.
'to keep' indicates the answer is hidden within the clue.
'standoff' can be found hidden inside 'insist and offer'.
(Other definitions for stand-off that I've seen before include "Deadlock (in match)" , "Rugby halfback" , "Impasse" , "Deadlock between equally matched opponents" , "Remain at a distance - while on feet?" .)