1: Can the contract to build construction be advertised and awarded to a private-sector firm before design is 100% complete when the design is occurring in house, like within the U.S. Army Corps of Engineers or Naval Facilities Engineering Command? 2: Can the contract to build construction be advertised and awarded to a private-sector firm before design is 100% complete when the design is to be provided by a contractor like AECOM?
What you are describing is a good fit for a Design Build contract. You can take a design at any phase from roughly 35% to 95% and award it as design build and the contractor will finish the design. That is the cleanest way to do it. FAR part 36 applies here particularly FAR 36.3.
it is possible you are describing another scenario where the government is going to continue the design work after award and you are looking at a Design Bid Build approach with the design not yet finished. Say for instance the design is complete except for the Electrical drawings and you plan to modify the contract to add the electrical work after the award. That would be unusual but there may be a way to do that, i.e. alert the bidders to that in the solicitation and indicate that the electrical will be a modification within 30 days of award. That's not as clean, but conceptually it could be done. You would be using Other Than Full and Open competition to award the Modification to add that work, so in that case FAR 6.3 would have information on the justification. Also, you want to talk with legal as this is not a standard approach.
But that aside, doing this Design Build is not a problem if the government stops in house designing when the design build contract is awarded.