The only thing in Tyson's defense really is that they definitely could not winter in Castle Black. The summer was unusually long which means that Winter is going to be unusually long (and how!). Stannis and his army effectively had no choice but to march on through. They were stymied then, really. They could not winter in Castle Black because that would take such an absolute age as to make the entire trip pointless. And marching onward was a doomed journey as well.
Frankly I don't see why they couldn't just take some of Shireen's blood. If it is that potent surely would only need some of the stuff for some ritual. Gendry's blood proved rather effective after all.