You have a right to become one of those people

As for Tennis, it could have a mixed bag effect on your body.
It should be pretty non-adaptive, because you're mixing up differing speeds and levels of accelleration, not just plodding along at the same speed and distance all the time, like steady state treadmill stuff.
However you will find that you're working one arm a lot and the other less so, so you're potentially getting lopsided fatigue to the body, if the muscles used to hit the ball are the majority of the time, on one side, because you use more muscle energies namely ATP and Glycogen, whereas anything designed to target muscles like weightlifting, should use proportionate amounts of such energies and create proportionate amounts of fatigue on both sides of the body, when the training is done correctly.
One thing you could do to correct this, is to hit against a ball machine and alternate the hand you attempt to hit with every shot, or change it after every 5-10 shots, so you then attempt to work the body more evenly.
Or you could use a two handed grip, hit the ball one way a few times, (say your bodyline facing one way), then use the opposing stance so the bodyline faces the other way, thus you should create a situation where the two handed shot technique, requires one leading arm and one for stability of the racket.
By changing your stance periodically every 5-10 shot attempts, you could then be hitting in a way, where the stabilising Arm and leading Arm, switch functions, hopefully enabling similar levels of emphasis, to be exerted on the muscles in both Arms, over the course of your session, plus some of the muscle in the Shoulders and and Chest area.
Tennis can be a way of certainly utilising Torso and Arm muscles, plus to some extent training the Legs, especially when you accellerate and effectively get a little sprint action in there, as coupled with the slower movement speeds, you are targeting both the slow and fast twitch muscle fibres, again in a way steady state Treadmill stuff can't, becasue FTMF's don't respond too well to "plodding along" type Cardio.
So perhaps thats a way to balance out what you're doing. So either of the outlined techniques give you ability to spead muscle emphasis, fatigue and energy use, more evenly across the body. The possibility that trying to hit with one stance, yields more successful ball strikes than the other stance, shouldn't matter too much, because by the time you could strike, the muscle movement and increased energy use and fatigue would be mostly utilised, so the actual ball strike / impact, would probably create little more, even if you have a more productive arm / stance.
Hopefully that all makes sense.
Plus the switching of the stances, could sharpen up fore and backhand game, or a left / righthanded game style too, giving you the chance to improve versatility.
30 minutes is enough really for people to do Cardio on a normal basis. Obviously for things like Marathons you need much more. You'll be fine doing the Tennis 3 times a week, at 30 minutes a time.
As for the Body consciousness in the Gym, I totally understand. Females are often more conscious and more critical of themselves and can be a lot more sensitive to change. I think it's possibly been a centuries old thing, where the desire to look acceptable to the of the species, has been inbuilt in many females, though I think it now extends, (to those who are conscious), to society in general, with regards to how you look in relation to how others think of how you look.
I can appreciate it must be difficult when you're in a Gym and you wonder if you are being glanced at, or observed in mirrors by some people, with a dissaproving look, as if you're not normal or have no right to be there, but I can tell you honestly, you are normal and you have as much of a right to be there, as anyone else does.
Again, any Gym or member(s), who predjudice against females with body issues, even if they make crass remarks, like you should stick to cardio and not do weights, are places or people that have no business being there.
It's your right to go to a Gym and train if you want to. Your attitude to others here has proven, that whatever you or others think of your exterior, you are on the inside a quite beautiful person and deserve to feel it in as many ways as possible.
I hope you do achieve your results, but I do think, that whilst some relevant eating patterns and a few sessions of Tennis per week, make this possible, without some kind of weight resistance work, they together may make an end goal, hard or unrealistic to completely achieve.
The Gym might seem like a daunting place, but peoples negative attitudes are there to be ignored and your right to be there is yours to be used and I think having those sort of facilities will be of additional benefit to you and maybe if you had someone else to go with, it wouldn't feel as daunting to you.
However though, GOOD LUCK and please do keep us informed of your efforts and changes. We DO want to help and by keeping us informed, we can do more for you and you have shown us you deserve our time and attention, so you ARE one of us now and we DO look after our own okay

Chin up and best wishes.
You're a great person

P.S. what promped the move from London to Texas, if that's not a nosey question. Sincere apologies if it is.