Well what do you know, using the latest drivers and firmware for onboard ASM 104X from stationdrivers, it actually seems to work with the Intensity Shuttle now. It would require further testing to see if I don't have any dropped frames or other issues while recording, but with the initial drivers I couldn't even get any video or audio out of them so this a considerable improvement. It's good to see that they're finally compatible now. So this means that you may be able to get it working after all, given the correct drivers.
Curtis, I suggest you check the firmware version and drivers of your ASmedia drivers. The drivers provided by Asus seem outdated, here are the ones I installed (keep in mind that it's not advised to use these drivers since they're not officially supported and could in the worst case mess up your system, but it's the only way to get up to date drivers):
http://www.station-drivers.com/page/asmedia.htmI installed the latest Version 1.16.2.0 drivers (top of the list). Those installed fine on my current setup (an Asus P8P67 LE motherboard). I also installed the Version 120816-02-02-06D firmware (the second on the list, not the first), which is somewhat more risky but it still worked for me. After installing those I actually got the Intensity Shuttle to work for the first time using the onboard Asmedia USB 3.0.
EDIT: that said, it still doesn't seem all that stable, I seem to have to restart media express after a while. In any case I wouldn't really recommend the U3S6 card if you have room for a single lane PCI 2.0 card like the Bufallo one. Blackmagic's support mentioned the PCI 1.0 4x cards sometimes had problems on PCIe 2.0 systems, so if you have a PCIe 2.0 1X slot available then I'd recommend getting the Bufallo PCIe card instead.