Years ago I used to use a Convergent Design box to go from FireWire out the old Sony DV decks to SDI into FCP. The convergent design box was called the HD-Connect-MI.
I think that thing only outputted SDI though, and I guess you don't have an SDI capture card, so you would need to further convert that. Blackmagic do an SDI to HDMI box for about £80.
In theory it should work, and it's all a digital connection so there 'shouldn't' be a drop in quality... plus the audio and video are always going down the same cables, so you shouldn't get any a/v mismatch.
Pretty expensive, inelegant solution... But a solution none the less.