In hindsight, limiting the motherboard options to those with Intel on-board graphics may have been unnecessary. It appears I need to go with a separate video card, anyway.
Problem #1: The Sony 1040Q projector can only accept 480i (320x240), and that resolution isn't natively supported by the computer.
My attempts to add 320x240 (with xrandr) have blown up so far... I can successfully add the option, but selecting it kills all video output (to both the DVI monitor and the VGA projector). I figured having the monitor running off of DVI, as opposed to sharing the projector's VGA-out via a splitter, would keep the 320x240 projector settings from affecting it, but no.
Problem #2: The Barco 708s projector requires a Component input. All of the video cards I've seen with TV-OUT use the s-video dongle. Some people say that is limited to 480i, others say it can support the full 1080p. I don't know which to believe... Also, I've yet to see an example of how that actually connects to 3-ch Component, as all the pictures I've found are s-video at both ends of the cable.
The DVI monitor and Barco projector both support 1280x1024, so I could easily run them as mirrored monitors (haven't had much success figuring out how to run dual displays that aren't mirrored ... every attempt loses essential controls on the primary) if only I could get the cabling sorted out.
Does anyone know of a video card that can support both 320x240 and Component output, so I can use it with either projector? It also needs to work with Linux Mint, of course, since that's what I went with.