The Mazda3 is a great choice. It drives like a sports sedan but costs like a compact. You can expect to pay about $14,000 for a solid used model, and running costs are low because they rarely break. To get the best price, shop late in the year or right at the end of the month when salespeople are desperate to hit their quotas.
Don't expect to put the whole $14,000 on a credit card. Most dealers limit card payments to protect their profit margins. Instead, use a card strategically for the down payment or the dealer fees, then pay the rest with cash or a cheap loan.
The 15-month window is a hard deadline, not a gift. If you still have a balance on your card when the 15 months end, the regular interest rate kicks in on whatever is left, which will quickly erase any deal you negotiated at the dealership.