If you are sure the PSU can deliver 5.1V constantly or more (but not higher than 5.5V) under load then most likely the USB cable may be another source for the undervoltage. Search for "micro usb 24awg 2c". The AWG24 wire gauge allows up to two ampere. (cf. here). And pick the shortest you can get. It may fix the issue.

However, from theory this may still not enough for an RPI3 (rated 2.5A).
You may try this, but do it on your own risk: Power the Rpi3 via GPIO headers. This bypasses the voltage regulator of the microusb port, so only use a PSU which has a stabilized output. Caution: Any mistake in wiring will fry your Rpi.

Get or make two DuPont wires with female endings, either AWG24 or AWG22 Cut them apart, thus you have two for 5V and two for GND at the GPIO header Properly attach the DuPont wires to the PSU, note the polarity Attach the female endings to the respective GPIO headers (use 2x 5V and 2x Ground) (https://pinout.xyz/#), again note the polarity Make sure the wires do not losen during gameplay