It was set to -10dBm at the time, and turning it down to -30dBm did seem to make it a lot more reliable.
But if it was a VSWR problem, wouldn't it behave like this with the output disconnected? That's as badly mismatched as it can get, but I can turn it up to +17dBm without triggering this problem, suggesting that it's quite happy with its own reflected power. (Maybe I should try it into a short, to see if the phase makes a difference?)
I had another thought, which was ESD. If I managed to accumulate a static charge, and discharged myself into the filter, this could have excited the first resonator and sent a brief pulse of RF back up the cable towards the generator's output. This wouldn't explain why turning the power down seemed to help, though, so I'm still puzzled.
Of course, being a second-hand unit from eBay, it might just have a minor fault.