I believe the problem in my case was due to an incorrect User class being specified. I had run
ipconfig /showclassid "Local Area Connection", which gave me the following output:
Windows IP Configuration
DHCPv4 Classes for Adapter "Local Area Connection":
DHCPv4 ClassID Name . . . . . . . . : Default Routing and Remote Access Class
DHCPv4 ClassID Description . . . . : User class for remote access clients
DHCPv4 ClassID Name . . . . . . . . : Default BOOTP Class
DHCPv4 ClassID Description . . . . : User class for BOOTP Clients
I thought, from this output, that I should have been using the Default BOOTP Class. After reconfiguring the options for the Default User Class, however, everything appears to be working now. Currently I have the option set in both the Microsoft Options vendor class as well as the Microsoft Windows 2000 vendor class.
It's interesting to note that a Wireshark capture still does not show the DHCP option being set. It seems that Wireshark displays this data under a Vendor-Specific Information "parent" option, but I can't be certain at this point.