Pilot reported a NMAC with another aircraft at a non-towered airport.

Narrative

Aircraft X (me and my instructor) are inbound to ZZZ on the LOC XX IAP. We were handed off to CTAF around 8 NM. We made two radio calls; stating that we were on 7 and 5 [mile] final for Runway XX for a low approach. Both times we hear Aircraft Y complaining that it is not a good idea to do straight-ins when the airport is busy (probably doesn't know we are IFR). As we reach around 2 mile final; traffic in front of Aircraft Y turns base to final; stating that they have us in sight. Aircraft Y says they cannot see us; then shortly afterwards they call turning base; putting them right in front of us at just a few hundred feet away and the same altitude. We arrest our descent and hold altitude to just let them descend away underneath us. Aircraft Y is too close to the plane in front at this point and says he must go around (since the runway is not clear); but knows we are above him so says he cannot climb. At this point we offset to the right of the runway to allow for him to climb and then we decide to just go missed and climb out on the published missed procedure.Our initial calls at 7 mile and 5 mile seem to have been received; evidenced by Aircraft Y going on frequency to complain that we shouldn't be doing a straight in after both calls. However; Aircraft Y's decision to turn base to final knowing that there was traffic inbound on final inside of 5 NM and not having that traffic in sight was extremely dangerous. The alternative was to just extend his downwind and turn base after we passed off his left wing.From our side; we should have gone missed as soon as Aircraft Y turned directly in front of us; rather than holding our altitude. We could have been above pattern altitude before crossing the departure end of the runway and been no factor.It's clear that there was a great deal of traffic at ZZZ. It was difficult to find a spot to get a radio call in; and there were many instances of peoples' transmissions being stepped on. The amount of traffic had Aircraft Y already on edge; they mentioned on the radio that ZZZ was 'a beehive' and that there were '20 people in the pattern'.I think some training on specifically how to handle inbound IFR traffic when you are VFR flying a pattern at a non-towered airport would be very helpful; and vice-versa.