Background Check - proof of unpaid part time

Thanks in advance to anyone who jumps in.

Here's my story:
I worked for a very small "non-profit" startup company for part time. I did some research and model building work. I did do what I said on resume without exaggeration. I didn't have a title there (since "company" very small and unpaid), so I used a title on resume that described my work there (in the line of , "modeling", "analyst", "researcher" etc.). The title thing maybe the only part that's questionable, anything else is fairly accurate. I worry about problems caused by exaggeration so I tried to be careful here. But again the title may be the place I exaggerated a bit.

I signed some paper work when I joined the "firm", but didn't really know too much about it. They said they're registered as a company, but I won't be surprised if they just say they are a school club. My future employer wants to verify this experience but unpaid employer won't prove anything for me. I didn't have a particularly good relationship with them but I did what I said I did.

Now the only things I can prove myself are things like interview emails before I joined, termination email, communication emails and work I did there. Do you think these will be enough? Will I fail the check?

I appreciate your help and apologize if I am being too paranoid. Any views and suggestions are greatly appreciated. Thanks.