Should probably go with the German greatsword. It is the only stabbing-centric greatsword, and stabbing is kind of the point of halfswording.
I must say, the longsword is also quite viable for halfswording. Yes it is short, but its speed on the stab is unparalleled. Not to mention it fills a unique niche for very short stabby polearms. The only other polearm in this space is the shortened spear, and there is something inherently more fun about halfswording.