After a good deal of experimenting today, it looks like IE10 and Firefox 24 both work if I click a link and call Page.Wait, but Chrome consistently returns from Page.Wait too fast, while the page redirect is still occurring. I've come up with a workaround method that I call after clicking any script-based link in our application. It seems to handle most of the random exceptions I'd been encountering, although only extended use will prove it out.
I'm not certain if this is a TestComplete/Chrome issue or if it's just a quirk of our web application. I'll attach the method here in case anyone else runs into a similar problem.