I have always wondered how in anglosaxon speaking countries, people think that "mint" is always the same as "mint", just because it has the same name. In fact, spearmint tastes as different from peppermint as thymian from oregano. Almost all cooking recipes I know of are meant for spearmint, except for some sweet applications. All mint tea I have encountered is made from peppermint, not from spearmint. So while you can use dried spearmint instead of fresh mint, using mint tea is a bad idea, unless yours happens to be an exception made from spearmint.
You could try finding out if a herbs seller has dried spearmint, but you must remember that it has less aroma than fresh spearmint. Also, dried mint does approximate the aroma of the fresh one when used as a herb, but when used in big quantities (you mention 1 cup) as a vegetable on its own right, the substitution is much more problematic, because juiciness and texture are much more different.
I don't know about the situation where you live, but spearmint isn't used much in Western countries, except maybe England, so it is seldom available at supermarkets and costs a lot there. A better source are Turkish grocery shops, where it is as common as parsley, and the price is comparable. If there are Turkish shops nearby, it is definitely worth trying to find it there.
For a longer term solution, it might be a good idea to grow your own spearmint in pots. The plant is quite unassuming and easy to care for, and a kitchen which smells of fragrant herbs is nicer than one which smells of frying grease or cleaning products.