In my opinion you're looking at the wrong problem. If you're burned out, you won't have spare energy for anything else, and however you approach it, fitness requires a minimum amount of energy to be sustainable, which you don't have. Technical solutions may lower the energy threshold, but they won't stick, as they won't lower the energy so much that you will stick to them long term, if you are mentally depleted. It's…
IMO the energy you expend doing exercise is coming from a completely different place than energy for knowledge work. On the contrary, doing exercise clears your mind and engages your body -- the opposite of sitting at a desk. In my experience you will feel your baseline "mental energy" also start to increase.
It's also only anecdata for me, but I do have hobbies that qualify as sports but after an exhausting day at work I want nothing less than to also exhaust my body and this has been a theme forever.
Also doing any kind of exercise in the morning is the worst and it doesn't clear my mind, it makes me want to get back to bed and the motivation to start work is at a complete low. Some people may well feel refreshed or good after exercising, for me it's the complete opposite - but I don't claim authority.