Two opinions here. Just two, really? There are currently 15 of us (after one integration).

First, if you are
skipping the sessions entirely, I think your T is obligated to charge you rather than your insurance for the no shows. Almost any doctor's office will have notifications on their intake forms that you are responsible for paying the entirety of any bill resulting from cancellation within 24 hours prior to your appointment. He has obviously notified you of his intent and obligation to not bill your insurance and I'm assuming has given you his policies as for how much time is required to avoid being billed. If this is what you're describing, than I think he is in the rights to follow his policies, although I know a number of clinicians might waive such instances if they were occasional, rather than a pattern. My T is very generous both time-wise and financially, because we are on a single case agreement that pays him very limited amounts and he always offers extra time and contact, but he is also very honest with the insurance (and I wouldn't have it any other way).
Second, if you are showing up to the sessions, but whichever part is fronting is just not able to talk, then I find his response more problematic. My T, also a Christian, makes sure to stress to me that there is never any need for performance. That I am valued whether or not I can make best use of the time we have. There have been alters who can't talk during session for long periods of time and are in obvious distress. He keeps reaching out to them in those cases and eventually they have all felt safe to begin talking, though it took a long period of time. He never once pressured them, but only encouraged them that he believed they could communicate and even if he couldn't understand, he would keep trying. I participated in another forum where a member's therapist told them they had a client who showed up and didn't speak in sessions for a year. He said through that experience, he learned how to wait, and the client learned she was worth waiting for.
Overall, I think we do have to take responsibility for the behavior of our alters. When one of my protectors gets angry and does some ***TRIGGERS*** punishing suicidal alters or others causing distress by what amounts to self-harm behaviors ***END TRIGGERS***, I can explain to my T and my H that I didn't "do" those things, but the fact is, it doesn't change the fact that a behavior that is happening within our system is causing pain and stress to others. It's not my job to really worry about my T, but I do always apologize for the distress, because despite it being out of my volitional control, I am part of the person who is causing problems. I'm not saying you need to apologize for your T, but I don't think we can entirely abdicate responsibility either, and should to the best of our ability work to reconcile parts who are interfering with recovery to the work that is being done.
What does this mean? In my case, discovering the reasons behind (especially a protector) part's behavior, negotiating permission to share those reasons with T and let him respond, try to arrange changes to the environment or the therapeutic process which will give that part room to experiment with building some trust, seeing whether it might be safer than they thought. While it doesn't always work perfectly, I would say that the behaviors that cropped up when we first started realizing we had something like DID 1.5 years ago have been reduced by at least 75%. And being shut down, having that part and other parts pushing to leave or quit or cancel has reduced at least 50%. It sometimes takes time and a lot of effort to work with a protector part. I'm assuming that is what you are dealing with, a part whose design is somewhat to avoid triggers that lead to overwhelming affect or memories (either biographical or just memories of emotion) in more vulnerable parts, which may influence you.
What if you tried to recognize the value of her contribution first? Like, I know you have done your best to keep me safe from getting hurt again or being overwhelmed and I appreciate how hard you have been working to do that. I know my protector has described it as a sort of "$#!+ job" or "cleaning up others' messes," so maybe it's possible she feels that way? When I stressed his importance to my survival all these years and especially since starting therapy, because he kept dangerous parts contained, T really acknowledged that even if if his methods weren't ideal, he did appreciate the times that my protector kept other parts from behaving in even more damaging and permanent ways. This helped the protective part to realize, "Oh, this guy gets it. He doesn't agree with me, but I don't need him to. As long as he gets that I'm trying to keep things safe and when I get angry, it's because others are getting out of control." He still disagrees with T fundamentally on a lot of levels, but he trusts that they have at least one common goal and feels safe backing off while we are IN therapy. It took a number of conversations (direct and indirect through me) to get to that point. But, I had to be willing to both be compassionate toward my protector's experience of us getting destabilized, accept his anger over it, and let it be communicated to my T, even though showing anger is very threatening to me.
I also wonder, as safe as it feels to be silent, whether it is not quite a lonely thing for her. Whether she wouldn't want to have a place where she could be safe by having someone be there and acknowledge/validate her feelings of fear and pain, to comfort other vulnerable parts she may be protecting, help them through the anger and grief that has been so overwhelming. The protector I am referring to hasn't really needed comfort or support himself, but has been provided so much relief having someone else "deal with" the kids when they are in pain. It wasn't an easy job for him and he doesn't have the "parenting" skills that would allow him to be comforting to them, more the skills of a bouncer. So, whether or not this part is actually mainly detached/avoidant, there may be a great deal of relief in breaking her silence, sharing her concerns with T, and allowing him to address them directly.
So, back to what I was saying before. If you are showing up to sessions (rather than skipping without canceling) and your T is being punative that you are not working enough in sessions, I might take issue with that. For an alter to show up in a session, even if they are too scared or otherwise unable/unwilling to talk, I think it should be honored, their trusting enough to be there. And, I think if the person is safe enough with that part for long enough, they will eventually find a way to communicate their concerns. The parts with the greatest trust damage will sometimes take a very, very long time to share. I had several hours (added up over many sessions) of therapy with one mute-child part before she tried to write stuff to T, and several more before she started talking, and now she talks within a few minutes of coming out. Sometimes it just takes a lot of time to build the sense of safety that certain parts need.
Sorry if none of this is applicable or doesn't resonate with you. I just thought my experiences might be helpful.