This depends on how quickly everything is integrated with US, but since the new US is/will be a Star Alliance (*A) carrier, and the OP is BMI Gold which Star Gold (*G) in *A. He will be entitled to use any *G lounge. This includes the AC MapleLeaf Lounge in the transborder terminal of YVR (pier E which is a very nice lounge self service bar overlooks the gates decent views for plane spotting), and all of the USAir Clubs.
Since the HP clubs in PHX are going to become USAir Clubs as a non-US/UA *G he and a guest will be able to use them even on a domestic or transborder flight. Regardless of your class of travel.
This is one of the nicest benefits to being *G, lounge access. US/UA *G have slightly different rules for lounge access. This rules do not apply if you have *G on any other carrier.
US and UA *G only have lounge access on an International Ticket. US currently requires you to be leaving North America to gain entry as a US *G to the USAir Clubs. UA on the other hand just requires you to be leaving the United States to gain access to the Red Carpet Clubs.
Based on what is published on *A website you should be able to get into any other carrier's *G lounge on a domestic ticket. Never tried it because I've always been on an Int. itnerary when I've flown through an terminal that has a *G lounge that is not a USAir Club or an RCC.
If you are on a paid Int. F or C ticket, and are not *G you still have lounge access. Int C tickets are not permitted a guest while Int F are.
Unfortunately both UA & US do not consider F on a transborder flight to be Int F or C.