Search for Quirk CBT. I think the core is open source and they have supporting apps for the various phone platforms. It isn't a general survey tool, but specifically tailored for CBT from what I understand.
I've not used it, so I can't endorse it one way or the other, but the author was interviewed on a podcast I listen to [1] and it seems like exactly what you are asking for. He went through a situation similar to what you are describing and created the project when he couldn't find software like what you are looking for.
Good luck.
[1] Changelog, Ep #345.
