We are going to need a little more information to help you. Im guessing the the OEM fob is the standard ford one with lock, unlock, panic, and trunk release buttons, but what is the remote start fob? Is it something you had installed at the dealer, or an install shop/yourself? Is it just a remote start module, or is it a whole aftermarket security system?
I'm guessing what is happening is that the remote start is not standard equipment, but feel free to correct me if I'm wrong. Therefore it only does what it is capable of because it is a separate unit. Think of it this way: on the OEM fob, when button A is pressed, the ECU does A, B, C, and D. The remote start is mainly to start your car from your fob. From what it sounds like, your headlights and taillights flash when you lock it and stay on when you unlock it correct? If the headlights flashing aren't a big deal to you, you could find the wire from the remote start module, cut it, and splice in the turn signal lights and the interior lights. I do not know what the current limits on the RS module are, and I do not know how much current each of those draw (I would imagine combined it would be less than the headlights), but I would not advise you to do anything until someone else gives us some numbers or confirms this works.