Unless you already have a contract calling for an "independent" CWI, then there is no (legal/moral/ethical) reason for one company not to have two of its own qualified employees doing the CM and the CWI jobs. Or one of its qualified employees doing both the CM and CWI jobs.
But ....
Is it really smart for the CM to try to do ANYTHING else during construction?
Literally, you would have him (her ?) focusing on one weld and one welder and one sheet of weld quals at a time, just when you NEED him to be managing the whole "strategy and tactics" and salesmanship and leadership roles of getting the project moving .... anticipating next week's problems while solving tommorrow's problems caused by yesterday's errors, and preventing today's errors from becoming tomorrow's problems.