Here, should is a modal auxiliary verb.
English word order is normally SVO (Subject-Verb-Object). Under some circumstances (including that phrase), the order changes to VSO, in a syntactic phenomenon called inversion. In this case, it seems to be triggered by wh-movement of who, the object of see, to the start of the phrase.
Note that it is the subject and the auxiliary verb that get flipped; the main verb will not move. To see why, I find it simplest to treat the auxiliary as the syntactic head, and analyze the main semantic verb "see..." as the object of should. (Aside: this phenomenon of verb-initial word order in questions is common to many European languages. However, it is no longer productive in English, and is now used almost exclusively with auxiliary verbs!)
The original phrase is:
who should she see but the Big Bad Wolf
Now find the underlying phrase that generates this using that transformation:
*she should (see (who but the Big Bad Wolf))
Now, to determine the lexical class of should, let's look at the un-transformed phrase. If the ungrammaticality of the untransformed phrase bothers you, it may help to replace some of the words:
she did (see (someone like the Big Bad Wolf))
In this case, should is in precisely the same lexical class as did would be.
Both are auxiliary verbs. Specifically, should is a modal auxiliary verb, because it is used to indicate modality.