Why not put an HDMI-to-DVI adapter on the output of the MonoPrice switch? I'd use something like MonoPrice's short "port-saver" HDMI extension cable at the adapter to take strain off the switch's output port, but those cost about $3. Then you can go HDMI out from two sources to the switch (maybe PS3 and HD-A3, assuming that you'll likely be loading a disc to use either source and can change the switch's setting at the same time) and HDMI-to-DVI-to-HDMI for the projector. The other two sources can go component through the 970 and out to the projector.
When I had an HDMI switch upstream of my 990, I found that it worked better to put the switch in front of the 970. That worked better because the 990 allows for a single DVI input to be assigned to more than one input. (I miss this feature on the Onkyo 885, which offers four HDMI ports but doesn't allow a port to be assigned to multiple inputs.) With the 970's fixed DVI inputs, I'd suggest arranging things so the 970's switching can automate things for you on the two most often-used HDMI sources. I'm guessing that those will be the PS3 and the DirecTV box. The HD-A3 (which isn't likely to be as good or convenient a DVD player as the PS3) could then go on the second port of the HDMI switch, thereby minimizing the number of times you have to go switch inputs on it. So that would be HDMI-to-DVI and optical from the PS3 to the 970, HDMI-to-DVI and coaxial from the DirecTV to the 970, optical from the HD-A3 to the 970, HDMI from the HD-A3 to the switch, DVI-to-HDMI from the 970 to the switch, and HDMI (-to-DVI-to-HDMI) from the switch to the projector, with the XBox 360 using optical and component to the 970 and component going from the 970 to the projector. In this scenario, the switch could live on the port tied to the 970 unless you wanted to watch an HD-DVD.