My friend's name is Allisan, how can I make it clear that her name is オーリサン and the honorific 'san' is not included and furthermore how can she introduce herself without sounding like a smug to everyone?
In self-introductions, 「さん」 is NEVER attached to the name in the first place. I have seen quite a few beginning Japanese-learners attaching it to their own names, but that is completely improper. They seem to mistakenly think that it is polite to attach 「さん」 to their own names when, in fact, you attach it only to other people's names.
Other people can address your friend as 「オーリサンさん」 with no problems whatsoever. It is the proper way. That it might sound funny is besides the point.
In self-introductions, your friend can say:
and no Japanese-speakers would think that the "san" part is an honorific. This, I can promise as a Japanese-speaker.
I have personally addressed people as 「イーサンさん」、「ネイサンさん」, etc. When pronounced with the right intonation, these do not sound nearly as funny as you might think.