Barbarossa is mid/late _twelfth_ century, which are rather different.
Mirliton do specific figures for this. They are found on their own website in their Crusades range (which confusingly also includes their 13th century Teutonic Knights).
Legio Heroica's Crusades range (and a few of their Feudal range which is mostly 13th century) should provide most of what you would want.
Some of Black Hat's Feudal range are suitable.
Donnington do a good Feudal range covering the 12th century.
Each has its pro-points so deciding on the 'best' is a very subjective thing. I'd be inclined to use a mixture but compatibility could be an issue.
In terms of size, I would say it goes from smallest Black Hat > Donnington > Mirliton > Legio Heroica. I haven't seen this Mirliton range in the flesh but going by the Muslims produced around the same time, the figures are probably bigger than their 13th century Communal range.
Black Hat and Donnington infantry would mix pretty well. The BH riders and horses are a bit smaller than Donnington's. Donnington mounted mix okay with Mirliton's Communal range but these newer Mirltions may be bigger.
Legio Heroica figures are lovely though I don't tend to mix them with other ranges. It isn't the height as much as the bulk and the style which I think makes them harder to mix.
There are other suitable figures e.g. Museum, Essex, Minifigs. If you ever come across any Alain Touller figures then these also go well with Donnington sized figures.
If you do want to do thirteenth century (which is fighting Frederick _II_ instead of the first), then my recommendations are still for the same ranges though I know that for this period Donnington, BH and Mirliton infantry, and Donnington and Mirliton mounted mix well.
EDIT: Waving Flag – those Mirliton figures you've linked to are splendidly painted, but they are from the fifteenth century range!
The link to the catalogue if for the 13th century figures, but see also link for the Barbarossa period
looks at some new terrain products, which use space age technology!