Windows VPN always disconnects after < 3 minutes, only from my network
Tremendous thank you to @Warner and @William for their suggestions. Ultimately it was William's answer which lead me to the final resolution. For anyone who comes looking, here's the deal.
After a ton of messing around trying to isolate the problem, I finally did as William suggested and pulled up my firewall logs. Not expecting to find anything interesting, I was surprised when I saw this line:
PPTP ALG rejected packet from xxxx to xxxx:1723
Knowing that PPTP is how this VPN is configured, I did some searching on the error. It turns out, other people have seen it as well. Specifically, people with my exact router, the D-Link DIR-655.
The solution, it turns out, is simple.
In the router's web administration interface, access the Advanced tab and click on Firewall Settings on the left-hand menu. In the section labeled "APPLICATION LEVEL GATEWAY (ALG) CONFIGURATION", uncheck the box for PPTP (optionally, also uncheck IPsec if your VPN uses that protocol.) Click "Save Settings" and tell the router to reboot. Voila!
Unfortunately, disabling these ALG options means certain advanced routing features will not work. For instance, the PPTP support is intended to allow multiple NAT'd clients to tunnel to the same VPN server simultaneously. That probably will not work if the box is cleared. However, if like me your VPN doesn't really work at all when the box is checked, you probably don't mind.
I am still unclear as to why I seem to recall having this issue previously with a totally different router, but I'm happy it is working nonetheless.
At a guess, there is a component of the VPN traffic that is required but being blocked (eg at a firewall) or lost, and causing the drop out. Check firewall logs if you have them for dropped packets. Double check the rules to ensure all necessary ports and protocols are enabled. You might also want to do some continuous route monitoring on your end to see if traffic is mis-directed after the VPN tunnel comes up. The "route print" command shows this info on Windows.