Many thanks to all who responded here. As Martin remembered faintly, the RTL-SDR dongle does not work well at low sample rates, and it gives considerably better reception to run it at (eg) 1.536 Msps and then resample it to 384 ksps than to use 384 ksps from the start.
Thanks to Tom for pointing out the ways of combining resampling and filtering. Was I supposed to know somehow that the rational resampler includes the necessary aliasing filters? That's not meant to be a sarcastic question - I'm trying to figure out where to look in the documentation for these sorts of things. I don't have GRC to hand, but I don't remember it being in the documentation for the block. The documentation for the rational_resampler_base_fff class sort of hints at it without saying as much, and it took me quite a while to figure out where to find even that.
At any rate, I've got the thing running in realtime, and performing quite well after replacing the crummy antenna provided with the dongle with a VHF TV antenna pointed in the right direction.
Thanks again,
Tom