Unfortunately you must now use idtravel.ba.com otherwise you may be unable to check-in.
It shouldn't ask for any info that you don't have access to.

There is a left luggage store at T5 where you can leave your bags but they do charge GBP 8 per item per 24 hrs or part thereof. You should be able to check-in as soon as you arrive ( early morning I assume ) and hopefully if seats are available drop your bags off at that time BUT you may be put to standby if you TELL them you are leaving the airport.

rgds