Alright, a workaround. Clearing cookies didn’t work. Switching browsers didn’t work. What did work was clearing cookies and then sending my traffic through a SOCKS proxy server on the west coast. After doing so I was able to get the fare that had been quoted to me, then snapped away.
The conspiracy-minded (e.g. me) will conclude that Virgin America keeps a list of IP addresses and, based on your fare-browsing activity, gradually increases prices in an effort to make you snap up a ticket before it gets any more expensive.
The more sane among you will recognize that the West coast is three hours behind me, and that the fare increase may be related to the time of day at which I was shopping.
Either way, jerking around your users in this manner — the way the entire airline industry does — is sleazy behavior. If you’re booking airfare and can send your packets through a western time zone, I recommend doing so.