Depends whether the Blackstaff choosing the heir means the bearer, or the artefact itself.
If Eb is making the call, I don't he would want Harry to have it, because putting the staff with someone Mother Winter could order to return it to her means the end of the Blackstaff as an office of the Council, and he believes in the need for that. It would also force the fight between the WC and Harry that he doesn't want to see happen, and it's an awful lot of temptation to mix up with the Knight mantle.
If the staff is making the call, a lot depends on whether IT wants to go back to Mother Winter. I think in Changes Harry perceived that the staff wanted to be used, and for all her malice, Mother Winter seems pretty self-disciplined about rarely actually exercising direct power. If its semi-sentience extends to having more fun being used for magical war crimes on the semi-regular on earth than it would being kept on a tight leash for centuries at a time in the deep NN, it's probably going to avoid the Winter Knight like the plague, lest he return it. In that case, Carlos is probably a good call as he's in opposition to the Queens' instrument.
If it operates on a lower level without that much if>then reasoning, it's probably going to see the Winter Knight as a preferred candidate if he's nearby. And even aside from the mantle, it would like Harry because here's a guy who gets into lots of scraps and will probably use a tool of violence.