British airways say:
Excess Baggage
If you have personal baggage that is in excess of your free allowance, British Airways will be able to carry your luggage in either of two ways
Excess Baggage
Unaccompanied Bags (not accepted for travel to/from USA.)
Excess baggage will incur charges, however you have the choice as to how much you pay, depending on how urgently you require your baggage.
If you want all the baggage to travel with you, then we will apply Excess Baggage charges - the amount is dependent on whether your allowance is based on the piece or the weight allowance.
Payment
Excess baggage can be paid for by credit card, cash or travellers cheques. If you know how much excess baggage you will be carrying you may also ask your travel agent or British Airways office to issue you with a Miscellaneous Charges Order.
Check in times
If you are planning to travel with excess baggage we suggest you check in at least 20 minutes earlier than the recommended check-in time.
My reading of that is that you'll be fine if you arrive another 20 minutes early, and you could even book your excess baggage ahead of time if you wanted. I've not heard of any excess baggage being refused to / from the USA and you're "only" talking 2 extra pieces. There is nothing in the text above that has any hint of "we may refuse the extra baggage if we're full" and that's the page that I would expect to find that warning on if it applied.
Beware the 32Kgs per piece limit, though. I've had to lighten a case at checkin; luckily I had one bag less than I was allowed and I was provided with a box by the checkin staff.
