I understand why you mention her weight gain leaving her feeling low and not her usual confident self hence perhaps more susceptible to men with a less than honourable agenda. Been there!
I also get what you're saying in that he's probably not a monster and he is known to your friend and her circle including you so it's hopefully nothing sinister, him wanting to move in with her and DC. If they get on they there's no reason a relationship couldn't develop.
However, I agree that it all sounds a bit convenient for him, parents want him out, your friend newly (ish) single and more likely to say 'yes' to him moving in. It would certainly save him getting his act together, finding regular work and a flat deposit.
There's not a lot you can do if she does agree to him moving in but I would encourage her to get to know him in a romantic capacity by dating, and absolutely not to let him move in, even on a 'temporary' basis, as it could be hard to get him out if things don't work out and not the best for her child.
Also, as a competent woman, doesn't she want an equal rather than someone who seems to have failed to launch to some extent? If she talks about her feelings regarding weight gain, remind her that even if she doesn't feel her best at the moment, this in no way lessens her value and that this man should be making the effort with her, not eying her up as a convenient landlady.