I study undergraduate philosophy. 

I wrote a paper (beyond the coursework) that presents a solution to one of the unsolved problems facing our conceptions of personal identity. I regard it as worth publishing, but I get that, as an undergrad, I may not have much of a sense of what qualifies a paper as "worth publishing." So I took a few precautions:

1. I read the recently published arguments, and the arguments that preceded them.
2. I critically thought about whether the argument could contribute to the field
3. I presented the argument to some people accomplished in fields that depend on logical thinking {a physicist, a lawyer,a mathematician}. The mathematician told me of a few useful concepts in math that I should incorporate into the argument, I studied them, incorporated them, and re-presented the paper to him, and he felt it was clear and logically sound.
4. I asked some academically successful people {two medical doctors and a Harvard mathematics graduate} for their opinion on it. They agreed with me; however, none of them had much experience publishing research.
5. I presented it to the prof.

The prof's responses seemed defensive. However, I get that I may have had a defensive bent that disposed me to regarding her criticism as her 'just responding defensively'. So I recalled the conversation to people who I believed would tell me that I had mischaracterized her responses, if they suspected that I had done so - and none of them did.

I also get that,

1. most professors -like anyone- want to make the most of their time, and that
2. most people usually see what they expect to see, and that
3. most professors probably would not expect an undergraduate paper to contain much worthwhile.

So, I suspect that most wouldn't want to read it, and that if a prof agreed to read it, an extra measure of prejudice would affect her assessment of it. 

I say all of that to say that I have reason to believe I should try and publish the paper, but I didn't succeed when I tried the most obvious path to doing so (presenting it to the prof), and I don't suspect that I'd have much success asking other profs to read it. 

So, how might I get a journal or a professor to give the paper a fair shake?