PA28 pilot reported an NMAC with a military trainer type aircraft in the pattern traffic of a non-towered airport requiring evasive action to avoid a collision.

Narrative

I was flying from ZZZ2 direct to ZZZ3 at 025 MSL with flight following by ZZZ Approach (Squawk XXXX). As I approached ZZZ from the southeast at 025; ZZZ Approach asked me to climb to 035 since along my route of flight I would expect Aircraft Y traffic returning to ZZZ at 3000 ft. MSL. I climbed to 3500 ft. MSL and turned on my nose taxi light (LED 11;000 lumens) as well as my wingtip landing lights that alternate on for 1 second each then the opposite wing (Wig Wag pattern of 11;000 lumens each light). As I neared ZZZ1 I observed 2 x aircraft in the pattern; ~180 degrees out from one another with one aircraft on the downwind and the other touching down on the runway. The aircraft performed 3 x tough and go landings. As I approached the departure end of the runway (my aircraft was approximately 2 nm north of the runway; and my track was essentially parallel to the runway); after the 3rd touch n go for the aircraft on the runway I observed that aircraft continue upwind; appearing to depart the airport. I looked over and saw the 2nd Aircraft Z depart upwind after the 3rd touch and go and takeoff upwind as well. By this point; the first Aircraft Y had continued upwind approximately 4-5 nm and I saw him in a right turn well ahead of my path. I looked back at the second Aircraft Z and saw him start his right turn within 1-2 mile of the departure end of the runway. I was approximately at the departure end of the runway at this point and still 2nm north of the runway. After the 2nd Aircraft Z had turned between 45-90 degrees of turn; I made a large wing rock to try and help the Aircraft Z pilots see my aircraft. A few seconds later with the Aircraft Z past 90 degrees of turn and still climbing; it became evident the Aircraft Z did not see me so I initiated a steep descent (thinking; this is it). The Aircraft Z passed right over the top of my aircraft as I passed through 3300 ft. MSL; and I estimate our vertical distance was 150-200 ft. I could clearly see the top of the Aircraft Z and heard it pass by. In the descent I bottomed out at 3200 ft. and called ZZZ Approach to inform them I had nearly had a midair collision. ZZZ Approach informed me they weren't talking to the Aircraft Z yet; and they gave me a frequency change to ZZZ Center. I tried checking in with ZZZ Center and no one replied. I returned to the ZZZ Approach frequency and told them I couldn't reach ZZZ Center; and asked if they had the callsign of the Aircraft Z I nearly collided with. They did not have the callsign (Possibly the Aircraft Z returned to ZZZ; VFR; Tower to Tower). ZZZ Approach asked me if I wanted them to mark the tapes; and I said affirmative. I returned to the ZZZ Center frequency and made contact.After landing at the destination; I called the squadron Duty Officer to find out if the Aircraft Z aircraft I nearly collided with was from that squadron. The Duty Officer said it was unlikely based on the type of training missions at that time of day; but he would check another squadron and pass the information I provided. Factors in this incident from my side; is that my Primary Flight Display had stopped working on this flight; which feeds altitude data to my transponder; so my aircraft was not transmitting Mode C data; just my Code and position. Additionally; I could have given the airfield a wider berth and not crossed just 500 ft. over the Class D airspace; especially since my transponder was sending altitude data. The Class D Tower would not have been able to know my altitude to pass to the second Aircraft Z; only a target 2 nm north; altitude unknown. I did not remember receiving any traffic alerts from my ATAS or TAS system and probably would not have with the Mode C data missing.