I came up with a stupid hack to work around this, though:
1. Open up a VNC (remote desktop) session on computer A, connected to computer B.
2. Make the VNC window full-screen.
3. Switch the KVM to computer B.
Now mouse A sends events to computer B, but the monitor is connected directly so there's no lag from the remote desktop.
(Incidentally, I looked into getting a modern KVM switch with DVI and USB support. But they seem a bit pricey: from $150 all the way up to $400. Because my new monitor only cost $400, this doesn't seem like a good investment--- I'm not that short on desk space yet.)