"I think we would struggle to afford him to live here fulltime but we would make it work by cutting back on a few things. DP thinks we should let him without needing to know as it's his business not ours."
Your husband is wrong on this matter. It does need to be discussed, because I would expect that this boy has not had a happy supportive home life since his mum died. Ages 12 to 16, that's rough.
My guess would be that whilst his stepdad accepted his wife and her son came as a package, when his wife died he felt considerably less responsibility towards her son. Probably regarded the boy as a duty, a chore, a restriction on his own life.
And now? He "has a new partner that has moved in", who may well be asking why she is expected to be stepmother to a boy who isn't even his son? Basically, this boy has become an encumbrance to his stepdad and he wants him gone.
Yes, it's also possible he has gone off the rails, but even then it would be in reaction to how he has been treated.
Where I diverge from you is that you "want to know why he was kicked out before agreeing to him staying". Your agreement is dependant on why he was kicked out. I think I would agree regardless, but want to explore with him what the last few years have been like so that I would know how best to support him.
So I am in the middle between your DP who agrees to him staying but wants to sweep the boys last few years under the carpet (NOT helpful to this boy), and yourself who will only agree once you've vetted why his dad kicked him out.
Oh, and if you do talk to the stepdad - be aware that he will have a vested interest in painting himself in the best light, and so will put all the responsibility for the breakdown in their relationship onto the boy. Do not believe him uncritically.