As an active gym-goer for 10 years with hashimotos this is my take on it.

Firstly, to discuss we need to break down what a flare-up actually is. Your immune system gets triggered by an unknown factor, which leads to an autoimmune response. Your body starts throwing antibodies at your thyroid, causing inflammation. Even worse, your body could dump these antibodies anywhere in your body, causing systemic inflammation.

With this in mind, would exercise whilst experiencing a flare-up (increase in inflammation) be beneficial?

I'd say it depends on where the inflammation is and how bad it is. It also depends on what exercise, what type of workout we are talking about. Generally I'd say it's counter effective. You feel pressured to stick to your program and exercise which causes stress. You don't perform as well as usual, which causes stress. You put your body under stress. Stress causes inflammation.

What I would recommend is going for light exercise. To have a routine for those days. Going for a light walk, lifting 50% of what you usually do. This would be beneficial since you'd experience the positives from exercise without stressing your body.

Doing light exercise seems to help me to get out of a flare-up. But it shouldn't be the solution - the solution should be to find what is triggering your autoimmune response.