United first class reward redemption question

I'm at ~320,000 miles with UA and I'd like to fly first class for the first and possibly only time in my life (with my wife). I've flown saver economy but I've never had to worry about being charged extra with booking on partners.

My ideal itinerary for this June/July is something like this:
DFW->SIN (2-3 weeks)
BKK->FCO (2-3 weeks)
FCO->DFW

Now my question is, since I need to book on United metal to not get charged extra, how do I book my BKK->FCO segment? If I book United first for DFW->SIN and Business on Thai+Lufthansa for BKK->FCO, is my roundtrip still 320k? How does the segment on non-United metal price out if I'm only using if for business? Also, I've been trying to find some saver first availability in June... is it really going to be nearly impossible to find 2 seats in first?