FIL is making the most sense. MIL has decided she doesn't want this to cost her much, and that is why she chose the college. That would be the cheapest place to go, if it even costs anything. (Kind of like getting a haircut at a barber college.) If it turns out to be really cheap, then she has no incentive to not drag this out for as long as possible. Though I think there is a limit to how much attention the school is going to want to invest in this enterprise. Leave it up to her to make whatever arrangements are to be made. Your husband shouldn't be calling her to find out the appointment time. MIL should be calling him when she has it finalized. This is her project. Let her be responsible.

You said above that your husband is anxious to get started and see if counseling "can help." What is it that he wants helped? That sounds like he is buying into his MIL's premise that something has to be worked out. MIL did not come up with this idea because she wants to know how to have a nicer relationship with you. She wants to have everybody acting more in accord with her will. That's her goal. She probably thinks you need to be straightened out. There's a good chance that the counseling student and her supervisor are going to eventually figure out what's going on here, if you hang back and let others do the talking.

If the MIL talks negatively about you, then you have the option to just not react. You jumping up and leaving the room could be just the sort of drama that your MIL can exploit. If she starts displaying her negativity toward you, that will show up where her head is at. You might want to consider just letting her yack away. There is power in refusing to react.

When you go home after a session, you and your husband can quietly decide whether or not to return for another session. This whole thing is starting to sound like a joke. Let your MIL play out whatever she will, and see what happens. Your husband may learn something from this.