I can understand your anger. And wanting to tell your feelings/thoughts in a letter and just give it at the end of a session.
I have done something similair. Last month my T told me she's pregnant and will go on leave from the start of April. She will be gone for about 5 months (she said, but what if she decides to not come back). And I'm getting a replacement T.
Your situation is even more unfair. She's going to only work at that other practise and just leaves all those other patients to look for another psychiatrist. And you're willing to pay for therapy out of your own pocket, but they don't accept that (which I think is weird).
My t actually changed to another workplace a year ago, but I could go with her, which I did. But if it would have been a clinic that wouldn't take insurrence, then I wouldn't have been able to keep seeing her. So at least I didn't had to leave her then.
But now she's going on leave for a long time. And I feel a lot of anger about it.

I have send her an angry email. I wrote that I think it would be better if I quit, because this is giving me a lot of extra worries and something about that leaving a client in the middle of treatment is one of the worst things a T can do (abuse being the worst).
I send it, I think the day after me session. I got an email back. It seemed like my email did got to her.
I kind of wanted to make her feel a fraction of what she make me feel. Mean, I know. But sometimes T's...
I send her an email back explaining some of the things from the other email. And at the next session we talked about it. I don't feel better.

Sorry for bothering you with my stuff. But I can understand how you feel. Though if I would be in your shoes, I would be even more angry than I'now with my T.

I think it would be good to give her a letter with whatever you want to tell her. Even if there's a lot of anger in it. And if you want to give it at the end of the session, do it. But do go back next week to talk about it. I don't think it would be good for you to keep all your anger or concerns for yourself. You've only about three months left with her.

For me it would be very hard to work on the things why I started therapy if I know that my T would be gone in a few months and it wasn't by my choice.
I feel this too. Why would I talk about my upcoming exams or starting college if she won't be there for me when those things will be happening. Since her ''news'' I only worry about what I will do when she's on leave and how I'll miss her and I feel abandeoned and I feel like I'm only work, that won't even cross her mind for a seconded when she's on leave.

I wish you all the best. I hope you can work something out and that you can keep seeing her. Maybe if you talk to her about it, she can talk to her boss or whatever?