Well, an obvious proposal would be to send out a drone and transmit the sonar ping from a different location.

You're right in that a broadcast is required, but I don't see why the broadcast should have to come from the sub itself.

The threat model isn't a torpedo, it's knowing where the enemy's subs are (and aren't). "There's a sub within half a mile of this point" is likely 90% as bad as "there's a sub exactly here" for most missions.

While water motors are not exactly my area of expertise, I would also expect that drone/torpedo motors are significantly louder than submarine engines and that launching or recovering a drone is itself a fairly noisy event that exposes the sub to traditional acoustic tracking.

Sure, all valid issues. It for sure would expose the general vicinity.

In terms of noise, keep in mind that submarines have all sorts of concessions already. A diesel-electric sub is quieter than a nuclear sub, but it has to surface every two days or so to get air to run its very noisy diesel generators.

I think a battery powered electric drone should be far quieter than any sub technology we have (steam/nuke is loud, diesel is louder). The quiet systems we have are all electric and therefore have limited energy capacity.

Wouldn't the sub itself also reflect the ping and give itself away?

Is detection of EM radiation really that easy? Broad parts of the EM spectrum are heavily attenuated by water - this is what makes underwater comms difficult.