Does the TX indicator on the LARK A2 light up for a few seconds and then turn off automatically?

This is normal behavior; the smart‑power‑saving feature is enabled. You can turn it on or off in the app. With smart power saving enabled, the TX indicator light will only stay on for 6 seconds when a button is pressed or when recording is started or stopped; at all other times, it will automatically turn off to conserve battery.