My plan is quite similar to yours. 2 months ago, I started exercising by swimming and very recently also weight lifting to get in shape and build some muscle. That's also great for practicing how to handle using men's locker rooms and showers without panicking. Cause then I also plan on getting into martial arts. Specifically karate. That was a huge passion for me in my teens and childhood after watching movies like "Mortal Combat", and I practiced it at a local organisation at ages 13 through 15, 2-3 times a week consistently and totally loved it. That was long before my transitioning and coming out, so I did that "as a girl" back then, and got up to 6th kyu grade which was green belt in that particular style of karate. After I quit due to my declining mental health, I always wanted to go back to it again someday when I felt better and could focus. That was 13 years ago now.
Now most of my transition is done, as I've been on T for almost 6 years, had top surgery 4 years ago and only have hysto left to go as far as I want to go with my transition. My mental health has gotten a lot better and keeps getting better, and I definitely want to go back to karate again! I just want to get more fit for it first, likely decrease my smoking a bit or a lot, also practice my balance and flexibility first as well, and also want to wait with starting until after I've had my hysto and that having healed properly. That op should likely happen end of this year or beginning of next, according to my guesstimations. So my thought is hopefully I can start karate by next fall, as the place/organisation I've been looking into only takes in new-comers at start of fall term each year.
I definitely like the idea of becoming somewhat of a professional fighter but for me it's more about seeing what my body can do, learning to control it well, also self defense is never a bad thing to know and karate is more focused on defense than offense, and I think it will be a great spiritual journey as well to grow more as a person, and then ofc just having lots of fun with it. Bonus would be meeting new friends as well and just having some activity to do in my never-ending amount of free-time.
I see it more as a "life accomplishment" for me and something I've been looking forward to as a part of my life post-transition when fully passing and not having worry about taking time off for surgeries, rather than it being part of my transition. But I do see me building muscle to look more masculine and doing something about my curves as part of my transition.
I'm not sure if or when I'd get into actual competing though. As far as I remember that wasn't really an option until you get up to like 6th or 5th kyu grades (it counts downwards) which takes years to reach even if you're a fast learner like me, and since it was so long ago for me I'm likely gonna have to start from scratch again. I'm glad my old gi and other gear still fits though. Also this will be another style of karate than the one I practiced in the past, due to a lack of availability where I live now, but seems to be quite similar. The one I used to practice was Shotokan, the one I'm looking into is Kyokushin.
As for competition though, I did do a little bit of that in the past when I was still in "girl mode" during my last 6 months of it or so, but it was mixed gender fighting. So, I mostly did that with cis men cause of them being very over-represented, and only occasionally with cis women as there were a lot fewer of them attending. Karate is not like perhaps most other forms of martial arts; men and women do get pretty much equal there as it's mostly not physical strength you're using. Sometimes women can have advantage even, cause of generally being more flexible. I had an advantage over several cis guys of my grade back then, despite being a tiny thing on estrogen, because I had better technique and greater lower body core strength. However, men more commonly choose that sport than women do.
So I think it's probably a great choice of sport for trans people, and makes me not worry at all about possibly competing with cis men in the future. However, competition in and of itself is not something I'm feeling super hyped about. It can be fun at times, but I'm not really a competitive person in general.
Also, I wouldn't have any issue to consider being out as trans at karate if that would seem like a good idea for whatever reason. I'm not stealth, out of choice, so. But then I wouldn't want to be treated as "weaker" or more sensitive than cis guys, so I might want to keep my mouth shut about being trans for that sake. But mostly on that note, I'll just take things as they come.