You mention that you find it difficult to work around his demand avoidance.
Could it be that the teacher is responding to exactly that? In my last place, there were a few potentially very talented kids who took the piss just because they had been told to do something - if they had a PE teacher who they wanted to please (a former professional athlete), they would do everything asked and more, but if they were in a bad mood or it was a woman taking the lesson, they'd at best walk or go off in the opposite direction, but mostly, refuse point blank until the other teacher came along and told them to stop being disrespectful, do another two laps/ten press ups and they'd get on with it.
Our PE teachers do a load of work on conditioning and functional fitness, even more so since lockdown has shown so much wastage of muscles, poor core strength and non existent aerobic capacity - so everybody does a wide variety of activities from dance/yoga to athletics, team sports and cardio. When it comes down it, though, they still have to get them to run at times, even if they don't particularly like getting out of breath or have wobbly legs. And other kids love running, whether sprinting, mid distance or cross country.
I despised PE as a kid. But I had market stall trainers, no sports bra and wasn't allowed to wear my glasses 'for safety'. Turned out I was actually rather good at hockey, though. I still completely absented myself from all PE for at least three years due to injuries from not being provided with acceptable standard kit and no allowance for what I now know to be a disability - I engaged again aged 15 when a gym was built and we could do things I actually liked. I found out later that having the right kit and learning about technique meant there were other things I really enjoyed - like, amazingly enough, running.
There's also Covid to take into account. A lot of school halls, gyms and every other available space are being used for suitably distanced assessments - they may have nowhere else to go but out on the field and need to run to keep warm.
I also wonder whether he's got more of a technique issue? If he's on the heavy side and is heel striking when running, taking long strides, he's putting a lot of force upon his muscles and joints and he's likely to find it horrible - but with a change to midfoot of forefoot striking, which many of the other kids might be doing instinctively, he may find that short paces with his weight landing down rather in front of him, that it's far more tolerable. The fact that he likes rugby would support that, as the quick direction changes make it more likely he's supporting his weight in a more mid/forefoot position. Some runners also say that breathing in on the left strike is less likely to give them a stitch than doing it on their right footstrike - might be purely coincidental, but I've heard plenty say it (and thinking about it, it's when I inhale).
It's certainly worth a discussion with the PE teacher and they could be an old school twat beasting who they see as the fat kid but perhaps more in terms of 'what is happening and what can you suggest' than 'stop bullying my child, they don't have to do what you tell them'?