I don't really use any of these weapons. The Morningstar is fantastic as a 2h weapon on foot (the only situation in which it has no penalty). The short reach for a 2her can be a blessing in a lot of situations, like sieges or just close scrums in the field. And of course guard crush and piercing damage.
As a 1h weapon, though, I think the Military Pick is clearly better on foot companions. I don't see what argument could be made otherwise. Now, the player is a different story because the player is actually capable of utilizing the versatility. If you don't have the slots for a dedicated 1h and a dedicated 2h, the Morningstar is a good all purpose weapon. A great one in fact.
The one place you could make a good argument for the Morningstar as a 1h weapon is on horseback. The speed bonus will increase the damage and the guard crush effectiveness. Weapon speed doesn't mean much on horseback, but reach does.
So to sum up, I would say:
Human, foot, dedicated 1h: Pick
Companion, foot, dedicated 1h: Pick
Human, foot, true 2h/1h usage: Morningstar
Companion, foot, true 2h/1h usage: N/A (can't do it)
Human, cav, dedicated 1h: Morningstar
Companion, cav, dedicated 1h: Morningstar
And bonus, even though not relevant to your question:
Either, foot, dedicated 2h: Morningstar could be the best dedicated 2h weapon in the game in certain situations, but will be outperformed by the big cutting 2hers (Great Bardiche) in most situations with some power strike.