Why does my Raspberry Pi 3 becomes unusably slow when plugged into an HDMI capture device?

I have a Hauppauge HD PVR 2 Gaming Edition HDMI video capture device that I use to record gaming screenshots and live footage. It works great with Xbox 360 over HDMI and with Roku/AppleTV (until you try to watch something with HDCP protection, heh), but when I connect my Raspberry Pi 3 to the device, the Pi itself becomes virtually unusable.

Symptoms: Audio extremely choppy, EmulationStation UI very slow to respond, and even when I close EmulationStation and just drop to the terminal it's terribly slow. You basically just have to blind type and wait a few seconds before you see the text appear on screen.

I am using the Pi for RetroPie, on top of Raspbian 8/Jessie. The Pi works great on the two TVs I've used it with (720p and 1080p), but it just completely lags out when connected to the HD PVR 2 GE.

When I use the default Raspbian HDMI settings, the Pi appears to the capture device in 1080p mode. I have tried forcing a lower resolution by tweaking the HDMI settings in config.txt, but while that did change the resolution it did not fix any of the lag. As I mentioned before, video passthrough works fine with other devices, so it's not lag in the capture software I'm seeing; the Pi itself is just behaving really strangely when plugged into the PVR.

Any ideas what could be causing the Pi to misbehave like this?

EDIT: And just to be clear, I'm not talking about the inherent lag that you get with any video capture device. There's always a slight delay between typing or moving a controller when using these things, but with the Pi it's like time stretches out... the audio only plays in half-second slices and when you type in the terminal the letters are rendered in batches.

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