Pilot flying fighter aircraft reported fuel gauge failure inflight. Pilot returned to departure airport and landed where the engine flamed out during taxi to ramp.

Narrative

Phase 1 test flight of a fighter aircraft. GPS equipped; but retained the Eastern Block style attitude indicator. ZZZ1 ATIS wind 090/13; 10 miles; -R; 055 SCT; 120 BKN;+20/+20; 30.06. Initial climb clearance was to 1;600 on a 350 heading. After takeoff cleared to 16;000'. Immediately after takeoff planned routing was not available with ATC providing vectors. Entered IMC at approximately 3;000 and remained IMC until reaching approximately 33;000'. The Eastern block attitude indicator operated in the roll axis differently than western style in that the horizon remained oriented with the aircraft and the aircraft symbol moved to indicate roll attitude. Shortly after entering IMC; significant special disorientation was experienced with intense concentration required to maintain the desired roll attitude and maintain assigned headings. The aircraft had neither roll trim nor a usable autopilot and was quite unstable in the roll axis. Pilot workload became intense in this situation while maintaining aircraft control; performing frequency changes and compliance with ATC issued altitudes and headings. The decision to abort the flight was delayed in anticipation of reaching VMC conditions on top per the weather forecast. As the climb was continued; workload did not decrease with altitude control becoming very difficult. It was during the climb in the mid-20;000' range that ZZZ [ATC] issued a phone number to call for 'Possible Pilot Deviation'. At one point ZZZ [ATC] became concerned about oxygen system operation; rightfully so! Cabin pressurization and oxygen system operation appeared to be nominal (oxygen mask was affixed). Meanwhile ATC vectors ('Toward ZZZ' [VOR]) continued the flightpath well north of the intended flight plan route. A westerly vector was received; VMC was encountered and visual contact with a large body of water was obtained. A southerly vector and descent were requested to descend below FL180; cancel IFR and proceed direct to ZZZ. Meanwhile; the aircraft Inverter failed which was unrecognized for a period of time. A major effect of this failure is that the fuel gauge ceased incrementing downward. By the time the failure was recognized (only indications are frozen fuel gauge and oil pressure gauge failure - no warning lights or Master Caution!) approximately 260 L of fuel had been consumed. A VMC descent was accomplished to return to ZZZ. In order to maintain VMC; cruise was performed at relatively low altitude (3;500 - 5;000' MSL). Actual fuel remaining was estimated by illumination of fuel sequencing lights and LOW FUEL (500L). A critical fuel state was realized; and [priority handling requested]. There were no suitable closer divert airfields; so continuation to ZZZ was decided. Cruise at a maximum conserve airspeed was accomplished; the aircraft landed; and engine flameout occurred during taxi to the chocks.