Yes, the DJI Mic Mini is compatible with the DJI Osmo Action 5 Pro.
Same OsmoAudio direct-connect ecosystem as DJI Mic 2 -- DJI Mic Mini's transmitters pair wirelessly straight to Osmo Action 5 Pro with no receiver and no physical port needed. DJI's own product page explicitly names Osmo Action 5 Pro in its direct-connect compatibility list.
“DJI Mic Mini supports the DJI OsmoAudio™ connection ecosystem. The transmitter supports direct connection to select DJI products without a receiver, including Osmo Mobile 8, Osmo Action 6, Osmo Nano, Osmo 360, Osmo Mobile 7P, Osmo Action 5 Pro, Osmo Action 4, and Osmo Pocket 3, delivering premium audio and simplifying your gear and workflow.”
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Confidence: high confidence