You will need some photo ID, but they are not as strict as other airlines. That doesn't mean they will let you go if you don't have it (because they won't) it just means that they accept a wider range of ID forms.
You will be asked to provide ID at check in if you're dropping a bag off as the check in agent will need to see it when you give then the bag.
I used to handle flyBe and we did accept bus passes for the elderly as these are issued by the local council they are considered official enough for domestic travel on BE. However, it's well over 18 months ago since I last did anything with BE, so the rules may have changed and I was at EDI, not BHD, so local rules may differ, of course.
Best idea is just to go and find your passport or, as previously suggested, contact the airline, tell them what you have and ask them if that's suitable!