I'm so sorry you are going through this! I don't have any answers for you on what to do, but I do know if you look up ways to slow down Hashimoto's through your diet, there is a lot of material out there. I am an endurance athlete (I do a lot of slow distance running) but by no means am I competitive. I was diagnosed with non-toxic multinodular goiter, Hashimoto's, and thyroid cancer earlier this year (thyroid cancer was probably due to chronic undiagnosed Hashimoto's for a very long time). I'm pretty sure my endurance training has covered up a lot of hypothyroidism symptoms for me, because I tend to have more energy on days I run and the lots of miles helped cover up my slowed metabolism (until my thyroid finally gave up and I gained weight while running lots of miles). I am 38 and probably had symptoms starting when I was 16, but definitely lots of symptoms from mid-20s on, and now I'm trying not to be mad at myself for not recognizing my thyroid problem much earlier.
My thoughts on your situation - Synthroid may help slow the progression of thyroid nodule and goiter growth. While your thyroid will probably still be under attack by your immune system, having your TSH in a lower range will prevent your thyroid from continuing to grow, especially if you have nodules (I had about eight - 1 cm sized nodules on my right thyroid lobe). Synthroid will also help prevent many of the hypothyroidism symptoms you are probably experiencing - and those are two good reasons to start taking thyroid hormone. Anything you can do to try to save what normal thyroid function you have is probably a good idea.
Additionally, there are a lot of foods you can try to eliminate that are thought to contribute to Hashimoto's symptoms, especially gluten, and you can try to modify your diet and see if that helps. My brother's girlfriend swore that changing her diet put her Hashimoto's into remission and her antibody levels decreased to an undetectable range, but by the time I was diagnosed I had a bigger (suspicious nodules on my thyroid) problem, so I did not try modifying my diet. You may need to do your own research on this, none of my doctors have mentioned modifying my diet with Hashimoto's treatment.
As for how endurance sports contributes to a thyroid problem -- there is a lot of information out on the internet indicating endurance athletes suffer from hypothyroidism at a higher rate than other people, but most of the information I can find are studies where people suddenly increase their mileage (in one case women running 14 miles a week increased to 30 miles a week and had a decrease in thyroid hormone), which is not the same as someone who gradually increases distance/has been doing endurance sports for years or decades. I'm still not sure how my running (I run between 30-50 miles a week) impacted my thyroid function, and whether that contributed to my multinodular goiter or if that was going to happen regardless of my activity level (I have run 30-50 miles a week for probably 15-16 years now). I am currently struggling to get my TSH back down to normal after my thyroidectomy completion surgery in August (it's December now), my last TSH was 11.10, so it's still quite high, and my running the last few months has been difficult (but I still managed to run 140 miles in December, so it's all relative I guess).
If hypothyroidism is fairly common among endurance athletes, there is a good chance you may know people involved in your sport who also have Hashimoto's - you might try to reach out to coaches or other athletes and ask for advice. Probably the best thing you could do would be to find a coach or someone in your sport who can give you training advice who understands how hypothyroidism and sports affect each other. Some people get a boost of thyroid activity following exercise, but maybe with endurance athletics we need more "recovery" time to let the stress hormones come back down.
Maybe there is a specialist in your area that you could see who treats athletes with thyroid problems? I'm not sure my endocrinologist or my ENT has any idea how my running affects my thyroid condition or vice versa, and any time I mentioned my running they just said to keep doing what I'm doing. (I ran 235 in October, when my TSH was between 16 and 12 and freeT4 was pretty low - so even more severe hypothyroidism can't stop me from running)
Anyway, I'm not sure if any of this advice will help you, but I just wanted to let you know that you are not the only one out there asking these questions - and sometimes it's difficult to just keep changing doctors until you can find one who wants to treat with both conventional medicine and changes to the diet.
If it were me, I'd take the Synthroid, try some diet changes, and find a coach or someone who can help you with a training plan and take in to account your hypothyroidism. My hypothyroidism/Hashimoto's has slowed my running speed way down, but I have every hope that once I get back to a "normal" level of thyroid hormone function (I've been using TSH throughout here, but what I'm really talking about is freeT4, freeT3, and actual thyroid hormone function in my cells throughout my body), I will be able to get my speed back up and keep on with my endurance running. This has been quite a year for me (Hashimoto's plus two thyroid surgeries) and I haven't given up on endurance sports yet.
PS sorry this response is so long... I spend a lot of time thinking about Hashimoto's, hypothyroidism, and endurance running, and have a lot of sympathy for your situation.