Check in staff are not "good" with immigration issues. They will simply look at timatic and try and match your situation with their information mechanically. They will never make a judgement call on whether US immigration will likely "care" about something or not. If there is doubt, they will refer to their own immigration specialists (there is a visa check desk at LHR, elsewhere this would likely be done by phone), and if there is still doubt they may choose to call US immigration authorities and ask for advice (note that such a call is their decision, not yours, so it is not something you can actually "ask").

I would think that USA and Green Cards are likely to correspond to well known cases for BA but their interpretations of what is safe or not may not match yours so in effect, if you are worried, you are always best off 1) checking timatic yourself to see what the check in agents will see and whether it helps your case, 2) bringing any written evidence confirming your entitlements or 3) if all else fails, being safe rather than sorry (e.g. renewing the passport a bit early).

At the end of the day, BA's check in staff responsibility is to ensure that they don't let on anyone who would likely to be refused entry at destination, especially in the US where this would come with a fine for the airline so don't expect them to take your word for it that you'll be ok if any situation doesn't look fully kosher on paper and US immigration authorities do not formally clear this in advance for BA.