I think part of the problem is LARP is generally assumed to be fantasty, since it's "role playing".
SCA is the "middle ground" since people have a "persona", and sometimes get called role players, but there's no magic, wizards, or other fantasy fluff.
The group you sound like you want is re-enactors or living history. These are the groups that seem to be focused on education, re-creation, and/or WMA inspired combat but generally not while wearing armor.
It's a tough situation, and I think your best bet may actually be to try to find some locals with the same interests in mind, and form your own group.