Hi Biggles so sorry your son has such a complex mental health problem. I know the depressive element of this illness can be severe. I think that's almost certainly the reason for your son's lack of motivation and negativity. I think quite honestly that it just isn't possible to have any understanding of the way depression affects us, unless you have experienced it yourself (sorry I don't mean just you, I mean anyone).
I suffer from intermittent depression which can be severe and it's a torment for sure. It drains us of any motivation to do anything, takes away any enjoyment in life, and so many other things. I know everyone's depression is different and so are affected differently and none of us knows how anyone else feels, but the thing about mental illness is that it takes away who "we are" - the "me" if you like. It can feel like a stranger has taken over your mind and that is very scary. It makes me feel ashamed even though I know that's not rational.
I think Latara has good suggestions, but I wonder if you've already suggested things like this. Your son's self esteem and confidence is probably very low at the moment and everything might be an effort, so thinking of going to a gym might just be a bridge too far just now. I wonder does he have any friends or has the illness meant that they have drifted away, which is often the case, especially with young people.
This will sound unhelpful, but I honestly think you may have to accept that there is not a lot you can do at the moment, and suggesting things for him to do isn't going to work. How does he spend his days I wonder - does he sleep a lot (most young people sleep in in the morning given the chance) - does he listen to music, play computer games, watch TV, I know they are all very sedentary pastimes, but really anything that helps to distract him and alleviate the boredom, help a little. He does need fresh air and exercise I know, but he may not have the motivation just now. He also needs to be with other young people. Does he have support from the CMHT - if so maybe he could be put in touch with other young people with MH problems.
There's a helpful website YoungMinds that is worth a look.