Thanks for comments above. I was just (trying - and stumbling) to highlight a number of issues:
i) the access/wait time - perhaps the cut-off of the first call was accidental - but it seems by no means unique, and the 20-30 minute wait time for both calls to the GoldLine could be considered rather long.
ii) while I appreciate that the call centre folks may not have all the information - or be able to provide such information, they should be able to refer passengers onto those who do know.
iii) IT - as ever, sucks. If the computer says no, it's no. Not really, it's the logic in the software. Just because the expiracy date is in the past, it (a) doesn't mean that the Green Card itself is invalid and (b) doesn't mean that entry to the US would be denied - the Green Card is just evidence of the I797 (Legal Permanent Residency) - which is valid indefinately until it is revoked.
Anyway, I have no doubt that we'll get back into the US - it's just the dodgy BA system that's the problem.