Could you not buy a cheap throwaway ticket ALG-EU, departing within 24 hours of arrival in ALG, and therefore tell BA you are doing Transit Without Visa?

Furthermore, I believe the BA flight to ALG departs from LGW so if you are arriving from USA at LHR, you have to collect your bags at LHR and re-check them (or not!) at LGW. Which would solve the checked bags problem.