You can quickly download and install the Straight Talk att.mvno APN (it's what allows your phone to use the cellular data network) by going to unlockit.co.nz from your iPhone. Just tap Create APN and choose Straight Talk.
It won't fix your MMS / picture messaging, though (unless you're sending a pic to someone with iMessage). For that you'd need to do the T-Mobile SIM card swap method to be able to access your phone's cellular data settings.
Carriers I've used: Sprint, T-Mobile, Cricket, AT&T, Verizon, Page Plus, Virgin Mobile, Movistar, AT&T Mexico, Telcel, Straight Talk, RingPlus.
Phones I've used: iPhone 6S, iPhone 5c, HTC One M9, Samsung Galaxy S3, LG G2, BlackBerry Storm, Nokia 820, Motorola Atrix HD, iPhone 4S, HTC One V, iPhone 4, Nokia Lumia 900, Nokia N9, Samsung Focus, Samsung Captivate, Nokia N900, HTC 8525, iPhone 3GS, iPhone, Nokia E61i, Siemens A56...