If I use non-government funds to pay for flight on non-U.S. carrier, can I then use government funds for other costs of that trip?
The text of the Federal Travel Regulations explicitly states:


  You are required by 49 U.S.C. 40118, commonly referred
  to as the “Fly America Act,” to use U.S. flag air carrier service for all air travel funded by the U.S. Government, except as provided in §§301-10.136 and 301-10.137 or when one of the following exceptions applies:
  
  [...]
  
  (g) When the costs of transportation are reimbursed in full by a third party, such as a foreign government, international agency, or other organization.


In this case, since you are not requesting federal funding for your air travel, you can use a non-foreign carrier. Clearly, foreign travel will most likely require lodging and per diem costs at non-US-owned venues—unless you happen to like staying at Holiday Inns and eating at McDonald's, Starbucks, or Pizza Hut. Moreover, the above portion of the regulations say that if improper air carriers are used, only the transportation costs are ruled unallowable. It does not talk about the per diem expenses for lodging, meals, and incidentals.