Your British Airways Visa has no foreign transaction fee, so using that while traveling internationally will save you 3% in transaction fees.

There is no hard limit on how many credit cards you can have open at once; it depends on your credit history. 30% of your credit score is determined by the average length of your credit history; meaning how long your credit card accounts and other loans have been open. Plan to keep at least your oldest 2 cards open long term. Closing cards can lower your credit score as it reduces the amount of your available credit.

If you want another rewards card, look at the cards with no foreign transaction fees. I recommend the Chase Sapphire card; there is currently a 50,000 point sign up bonus and its points can be transferred to BA, Continental, Korean, Hyatt, Marriott, Priority Club, and Amtrak. This card requires spending $3,000 in 3 months. Once you've met the minimum spend requirement, find another card with a bonus that will help your personal travel goals.