Well if you get one without build quality issues, the Thrunite Neutron 1C is under 4" and will put out over 300 lumens on a 16340. My Neutron 1a seems to put out over 200 lumens on a nimh and I have little doubt that it will do over 300 lumens on a 14500 and the 1C seems to use the same head. I would like my 1a if it didn't have the darn wobbly head. With a $500 budget, you could buy 9 of them.
old4570 reported a (probably fake) Solarforce L2m XML T6 1000 Lumen 5-Mode Flashlight (1*16340) in another thread. According to Kaidomain, it's 10.60cm long.
Xeno E03 with XML on 14500. Costs about $30 for cool white emitter. 95 mm long and 490 lumens on high with the XML driven at about 1.5 amps.