Flight Instructor reported a NMAC on downwind in the traffic pattern at a non-towered airport.

Narrative

We were established on the left downwind for runway 29 at LMO. There were at least six other aircraft established in the traffic pattern at LMO. Another aircraft (Aircraft Y) reported over CTAF that he planned to overfly midfield and enter the traffic pattern in the downwind. Another aircraft in the pattern advised Aircraft Y that there were already three other aircraft established in the downwind and advised against their plan to overfly midfield to join the downwind. Aircraft Y proceeded to overfly the field at the traffic pattern altitude of 6100' MSL and join the downwind directly parallel to our current position in the downwind with less than 300' of horizontal separation. We decided to take a right turn to the south and leave the downwind to avoid Aircraft Y. If we had continued downwind as normal; Aircraft Y would have hit us. After we left the downwind; the Aircraft reported over CTAF that they were established on the downwind for 29 without ever acknowledging the near miss or us needing to deviate to avoid him. We then left LMO rather than attempting to get re-established in the traffic pattern and headed back to our base airport.