Looking through the lens of an American Army unit, a battalion command lance should reasonably be composed of the Battalion Commander (BC), Executive Officer (XO), Operations Officer (Ops), and a specialist of some kind. The latter could be a 'bodyguard', 'runner', scout of some kind, or perhaps a liaison for close air-support or artillery/fire-support (look up what an S-3 Air does for an example). The other staff functions, notably administration (S-1), intelligence (S-2), supply (S-4) and other sundry duties, do not have specific battlefield roles requiring BattleMechs or tanks.
Realistically, given a structure as seen above, it would be exceedingly rare to see a whole battalion headquarters lance together on the field at the same time. Using operations of the 11th ACR (in the guise of the Opfor/32nd Guards MRR) in the late 1990s at the National Training Center as a template, the BC would be leading the battalion main-effort, the XO would be leading the Jump Tactical Operations Center (imagine a stripped down version of the primary operations center) or the supporting effort, the Ops would be observing a critical part of the battlefield or involved in the secondary/supporting or diversionary efforts, and the supernumerary (fourth member) would either be protecting one of the principals or conducting their own specific duties.
But, you know, the BTU is a place where significant portions of available combat power assemble at a given point and charge at the other guy like reenactments of American Civil War or Napoleonic era battles. See in that light, having the BC supported by three bruisers as a personal defense detachment par-excellence is not strictly a bad thing. That is not how I run my universe any more, though (but in my younger years, that is exactly how my campaigns rolled, and I made the most out of BattleForce in doing so).