Can I register a ManCo first while waiting for the fund formation process to start, then roll it in to the fund's structure later with updated documents?

The ManCo entity should generally be registered at the same time as the Fund's other entities, using specific (and not generic) legal agreements. Trying to roll such an entity into the Fund's structure later on will slow things down, and most likely it will be more efficient and cost effective to discard it (and set up a new entity) than to roll it in.

TL:DR: It’s not efficient to do it this way. A Fund's entities should generally all be formed at the same time.