My usual advice to anyone whose doctor refuses to order a full thyroid panel, even after the patient has provided information on how important it is, is to find a different doctor. I understand that your doctor is very close to where you live and is in the network of your insurance. Insurance companies, however, rarely only have one doctor in their network, so there's bound to be someone else you could see to get adequate testing/treatment for your thyroid. That doctor could be a primary care doctor, internist or another kind of doctor. Another doctor might not be as handy as 1/2 mile up the road but if you get better treatment, it's well worth traveling further to see someone else. If you're not willing/able to do that, you're right, you won't make much progress unless you're willing to change some of your own habits to try to make improvements on your own.
Green vegetables aren't the only ones that have fiber in them, nor are vegetables the only foods that contain fiber. Vegetables come in all colors of the rainbow and each of them has a certain amount of fiber, though some have more than others plus they each have different nutrients we need. That's where a nutritionist might come in handy. There are a lot of ideas you can use to acquire a taste for new veggies.
I'm not much of a vegetable eater either, but I've done some experimenting with herbs/spices, steaming, roasting, etc and figured out ways to fix a variety of veggies I never used to eat, that make them quite tasty.
There are several supplements you could try that might help boost your thyroid, but of course, it would be helpful to know what all your thyroid levels are, along with ferritin, vitamin D, B-12, cortisol, etc.
In addition, there are places from which we can order our own thyroid panels at reasonable prices (less than $100 for TSH, Free T4 and Free T3) available in most states, without a doctors order. This is an option I recommend when one can't get their doctor to order the tests. Many times if one can get the tests done and show the doctor that levels are too low, they can get the treatment they need. If the doctor still insists that because levels are in the "normal" range, the only options are to find a different doctor, try to self-treat (not recommended) or stay sick (also not recommended).
You've posted in the Thyroid Disorders community in the past, and reading over the thread you posted it would appear that your thyroid testing/treatment is not optimal, or even close to adequate, for that matter. I know you've posted here, that your thyroid is overactive, but you posted in the Thyroid Disorders community that you are on Levothyroxine, which means it's actually under active, after being overactive. Without adequated testing and treatment for any thyroid disorder, it's likely one will have a difficult time with weight.
The thyroid controls metabolism, along with many other body functions and we must have optimal thyroid function in order for everything to work the way it's supposed to.
Of course, one can lose weight with the Alli diet because it cuts out all fat; the problem is that our bodies need some fat in order to be healthy. As the others have stated, we also need the nutrients we can't get if we eliminate vegetables and other foods that contain fiber.
Herbs and spices can go a long way toward making foods we don't care for taste good.
I'm a boomer, with hypothyroidism, too and one of my previous doctors actually prescribed Xenical, which is the prescription strength of Orlistat, the active ingredient in Alli, along with the diet to me some years ago. Prescription Xenical was expensive so I got Alli to try instead. I lost some weight rather quickly, but it wasn't without a cost as it can be quite "messy" and somewhat "surprising" if one isn't meticulous about the amount of fat consumed. I only stayed on it a couple of weeks... I've begun to see a holistic Nurse Practioner who has helped me with my thyroid, diet, and supplementation. The main thing was balancing my diet with adequate protein, fiber, fat, carbs, etc and as long as I stay away from the bread, sugar, dairy, etc, I can lose weight slowly, but surely.
As has been noted, sugar is an inflammatory and exacerbates pain of all kinds, including that from RA, hypothyroidism, etc. Along with sugar, anything that is converted to sugar quickly in the blood, such as things made with white flour, white rice, etc do the same thing and should be eliminated from the diet.
Your original question indicates that you can't lose weight, and in the end, you say you're losing weight, so I'm not sure how we can really help you.
We don't, typically, advocate diet pills for weight loss. We advocate healthy eating and exercise. But first, thyroid function must be optimal before one will be able to accomplish much of anything.
I understand that RA makes it difficult to exercise, but I know from friends/relatives that have RA, most doctors recommend as much exercise as possible because not using the muscles and joints can make them stiffer and less usable over time. Some exercise, if possible, is better than none. Of course, one should always consult their doctor.
I'd check the contents of your breakfast; what is its carb count? And drop the diet pill, I've never heard of even one of them that works.