If we are talking about command mechs, why not get a cyclops since it has battle computer. I can see why the exterminator is nice, being a fast mech and all, but we would get more with a cyclops in tow.
They do have 11-A about this time with a potential for a DC variant as well.
The Cyclops does indeed have a Battle Computer to give a +2 to initiative if the person driving it is the mission CO. The 3 highest ranked lance commanders are Midnight, Dancer and Wolverine, all of which are geared towards faster mechs as part of their preferred lance roles. Wolverine when present leads Bravo (Fight), with Midnight and Dancer in Alpha (Scout). Skaraborg is the company commander over Charlie (Defend) and I wish you luck in getting him to swap out an Awesome or the Turkina for a Cyclops.
Crusher is currently the company commander for Bravo until Wolverine returns, so that has some possibilities and not too far off Crusher's standard movement profile.
While less mechanically optimal, a Command Console provides the same bonus. There are stock versions of Mechs with Command Consoles including:
Battlemaster
Atlas
Orion
Awesome
Cyclops
Marauder
Zeus
Thug (impressively nasty canon variant)
Grand Dragon
As for the Exterminator preference, two of the three lances that have expressed an interest are both in Alpha Company, so speed and maneuverability are important given it's role. Bravo is supposed to be the next most mobile and so some of it is personal preference. There's also the sheer imposing factor of stealth equipped heavy headhunters on the battlefield, especially in 3052. Now, keeping it running and out of most opponents crosshairs might be a different matter.