Most private contractors for the government ARE already like that and getting a profit. 
You may have some non-profits doing specialty work for the feds, but much of that was done via grants, and the whole grants thing, and the rest, have recently been targeted by 45 (and Muskrat's DOGEshit), and have been or are being purged
Contracting Officers have different types of contracts that they can solicit for - like "fixed price" or "cost-plus" (the latter with extra financial incentives, often to have the work done by a certain date and/or for specialized fields needing an expertise that is difficult to find).
Even with the "fixed price" contracts, the prime is going to make sure they make a profit somewhere (including hiring lesser-paid workers and/or using cheaper parts, etc).
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