I'm still hammering this out, but it works as-is:

@echo off
rem Simple script to map a drive using the new IP provided.
echo Enter final octet:
set /p OCTET=

net use z: \\10.0.100.%OCTET%\c$ /user:inside\<username> <password>

I'd like the octet entry and the echo to be on the same line, but aside from that, working as advertised.

A way to test whether your domain login credentials are working is via net use on the commandline like so:

c:\> net use \\<computername>\IPC$ /user:<domain>/<username> *

The * forces it to ask for your password. Otherwise, you can replace * with your password to do it all at once. Also, net use supports smartcard authentication, etc.

Thanks to Scott Morrison (a.k.a. "Plaid") for the tip!