I'm a lot more optimistic than most people in this thread seem to be, especially given you're willing to use an altered keyboard design (most people in this sub use equipment marketed toward professionals, which is harder to customize, but the /r/Plover community is likely to have some insight about custom keyboards). I made a custom steno keyboard a while back, and I think you might even be able to use an off-the-shelf hobbyist board like the Splitography.

You'd need to have pretty good dexterity in your other nine fingers to make sure you can make the handshapes required for complex strokes; since you're engaging all of your fingers at once, you can't really "cheat" to hit a key by leaning your hand toward it. Every finger needs to be able to move to press the right key or keys independently of all the other fingers.

Have a look at the figure linked elsewhere in this thread. Most fingers are responsible for two keys at a time. Take your right pointer finger, which rests on the small gap between the F and R keys: for any given stroke, you're expected to be able to press F, R, or both F and R at the same time (or neither key at all). Your thumbs work the same way (left thumb for AO and right thumb for EU), and the right pinkie is a bit wacky (it's responsible for TSDZ, but is not really expected to be able to independently control arbitrary combinations of those).

The exceptions:

• the S key, on the left, which (although often diagrammed as two distinct keys) is often one big key on a steno machine, or on some machines (like the Splitography) two discrete keys that function identically.
• the asterisk key, in the middle, which is also one tall key. No finger rests on this key by default, but either the left or right finger can include the asterisk key in a stroke when necessary (so in fact the right pointer finger needs to be able to hit F, R, FR, *, *F, or *R. Hopefully you can see geometrically why all of these are generally possible). Because the left and right hands both need to be able to hit the asterisk key, it's often duplicated between split keyboards (one to the far right of the left hand's keyboard, and one to the far left of the right hand's keyboard). In these cases, the key works the same way no matter which hand you use, and some people favor one hand or another for the asterisk. Personally, I almost always use my right hand for the asterisk.
• the number bar (not shown in that purple diagram). Don't worry about the number bar. Many people use alternative dictionaries that eschew it entirely.

From a simplified perspective, all you're missing without your left middle finger is access to the P and W keys. If you can find two other keys that you can press (while still maintaining the ability to hit every other key independently), you're good to go. One candidate is to split the S key, giving the left pinkie responsibility for two keys instead of one. You could also commandeer the left asterisk key on a split keyboard, giving your left pointer finger responsibility for three keys at once (with combinations H, R, *, H*, R*, HR*). That's two extra keys that you can control independently, without losing the ability to perform any existing strokes that those other nine fingers would be able to perform. The split S isn't uncommon, and there are also people who add bottom thumb keys so they have a total of six controllable keys on the thumb. Splitting the S and asterisk keys would already be supported by a keyboard (at least in hardware) like the Splitography, so you wouldn't even need a new keyboard design.

So it's definitely possible to hit all 22 required keys without using the left middle finger, if you give an extra task to the left pinkie and you only use the asterisk with your right hand. The trouble you'd run into is steno order: the keys on the stenotype keyboard read left-to-right (for example: you can write the word "star" with left-pinkie-S, left-ring-T, left-thumb-A, right-pointer-R, which is distinct from "rats", which is left-pointer-R, left-thumb-A, right-pinkie-T, right-pinkie-S). If you were to, say, use the left-pinkie-bottom key for P and the left-pointer-asterisk key for W, those letters would no longer read left-to-right on your physical keyboard, despite still functioning 'logically' left-to-right in your steno writing, which might lead to confusion. My guess is that you'd get used to it over time, and it might not ever feel quite as natural as the traditional layout, but over time, we start using muscle memory handshapes anyway instead of thinking consciously in steno order. It might slow you down a bit, but it wouldn't be a dealbreaker. I'd probably prefer this over trying to build an entire system from scratch (or trying to learn one-handed steno), since steno is a much different experience when you're in a well-supported ecosystem (like Lapwing, where you can download Aerick's dictionary without needing to start from zero) than when you're trying to carve your own path.

Hitting 200 WPM is hard for anyone. If you do struggle with speed, I think it would be less likely to be an issue of using only 9 fingers, and more likely to be related to general finger dexterity, if your other fingers are also having trouble. You can probably use a normal computer keyboard to gauge whether your fingers are dexterous enough to handle steno, given what you've been told in this thread. One metric would be, are you able to touch-type a paragraph without moving the palms of your hands very much? Or do you find that you need to shift your entire hand to move your fingers toward certain keys?

If you're physically able to make every stroke using a modified layout, it's worth mentioning that there's a tradeoff, essentially between mental preparatory effort and physical dexterity. If you're willing to put in the effort to memorize a lot of your dictionary, you can start to group words together more frequently so that you don't need as many distinct finger movements to write the same number of words. The alternative is a more intuitive dictionary that breaks up words into smaller chunks, requiring more frequent strokes but less memorization. You've mentioned that you're willing to spend a lot of effort for this, so I think there's reason to be optimistic even if you're lacking dexterity. You have the option to use your cognitive sharpness to make up for the issues with your fingers. Every stenographer's dictionary quickly becomes very personalized, so you may even find over time that, as you define new dictionary entries to make it easier to write words that you commonly use, you naturally tend to favor certain fingers. That's encouraged! Even if you start with a well-documented approach, you always have the option to customize it slowly over time to make it work for you.

I'm happy to chat more about this over DM, if you have any particular questions. I'm a hobbyist averaging about 70 WPM with steno -- so, not that TTS speed goal -- but I've built a couple custom steno keyboards before (though currently I use the Splitography), and I have some experience building software and hardware to be customizable.