Genuine batteries seem to last in the 5-10 year range. Replacing the battery with the right one should give you that much again. Even if your non rechargeable is lasting a year before finally dying, you're a long way behind after saving your initial $2. If you get a month making you happy, you'll be a long long way behind.
Also it won't "melt". Depending on the battery chemistry it may leak something corrosive potentially damaging the electronics.
Your money, your car and your key though, so good luck.