Wireless Time Synchronization Project
Thank you for bringing up an interesting, practical problem to solve! You have some interesting ideas in the comments, but here is the approach I would take.

Use Bluetooth Low Energy for your radio protocol. Adequate range, low power consumption for battery operation, software platform options from C to Javascript to Arduino, and a choice of packaging from breadboard-ready PCBs, modules, to bare chips. The Bluetooth chips have more than adequate CPU processing power and speed, and you won't need a separate MCU on either end.

All BLE chips have an internal clock with about a 30 microsecond resolution (they run at 32.768 kHz), which can drive an internal counter. You don't need to match these counters to actual time-of-day; they will report ticks since the device was booted up. When the sensor in the mitt is triggered, record the internal mitt clock time. When the sensor in the base is triggered, record the internal base clock time.

I'd make both devices BLE peripherals, reporting to an app on a smartphone that connects to both of them as a BLE central.  When the app gets a report from both peripherals, note those two times. You don't have to worry about the latency of the transmission delay -- this only affects how quickly you get the report.

Now you just need a calibration approach. At the beginning of the game, whack the mitt against the base, so both sensors are triggered at the same time. In the app, observe the two clock readings, and memorize the difference. This is the &quot;offset&quot; when the sensors are triggered simultaneously. By subtracting it, you can tell the relative delay between the two events.

Will the clocks stay synchronized well enough? For typical +/- 50 ppm BLE 32.768 kHz clock crystals, this represents a potential error of +/- 0.18 seconds over one hour. So it might be best to repeat the calibration before each inning, which also tests that everything is working.