I'll just come out and say it. BYU would benefit in no way by playing Washington. All season long, all we've heard is a constant random repeat of these phrases:

"BYU hasn't played anyone."
"BYU's schedule is filled with a bunch of (FCS, Jr. college, high school) teams."
"BYU will be exposed as weak once they play (Houston, Boise St., P5 Team)."
"BYU is overrated. They should never be considered for (NY6, CFP)."

If BYU were to actually play Washington, one of the following things would happen:

1. If BYU lost, everyone else gets their confirmation bias fed that BYU sucked all along, and all we hear is "BYU was overrated. As soon as you play a big boy you fold like paper."
2. (More Likely) If BYU beats Washington, the goal posts move again, just like they did for Houston and Boise St. when the naysayers were so certain BYU would lose. They will continue being naysayers, only now we'll hear "Washington sucks this year. If BYU played a real PAC-12 team like Oregon or USC, BYU would get trounced."

It would be the same thing BYU fans have heard ad nauseam all year long. The PAC-12 will always be the PAC-12, in that they will always be the least secure of all P5 conferences, claiming greatness, while chasing the coattails of the B1G, SEC and ACC.

In no way shape or form should BYU schedule Washington. It's mostly a no-win scenario. Best case is BYU buries Washington like they have most teams this year, and the status quo of "BYU hasn't played a good team" continues.