Hi sunbrother. You're right that I did stop automating the bypasses via the 'Main' option, yes. And this solved the problem of clicks in the audio in the bounce down. But it didn't solve the phasing issue, unfortunately.
I think the phasing issue may simply have been bad luck with the timing of the bounce: the CPU, for whatever reason, was underperforming and that led to artifacts.
Now there was actually one more thing I did: that project has a snare-attack aux channel, and I changed the Logic Enveloper lookahead time from 2ms to 1ms. After bouncing down, the phasing was gone - so I thought perhaps the lookahead time had been the culprit. However, to check this, I did another bounce and reverted the lookahead back to 2ms, and still I had just as clean a bounce, so I don't think that was it. (And, in any case, lookahead settings might account for phasing on the snare, but not the whole kit - and it was the whole kit that sounded unnaturally thin, not just the snare).
The only difference, therefore, was moving the sequencing of channel strips around to conform with all the other projects I'm working on, but this seems very unlikely as the deciding factor. But you never know! Complex software creates all kinds of unforeseen relationships between seemingly unrelated things, so I thought I'd mention it anyway.
Cheers for all your input on this issue, sunbrother. I appreciate it. And you have persuaded me to no approach 'bypass' on my plug-ins with caution and instead look for alternatives such as 0% on the mix settings. A great tip (a tip of the hat to SoSpiro here, too) 🙂