To list or not to list...HELP!

Hey guys, I need some advice. I'm a 2nd-year undergrad in the UK and currently engaged on a two-month consulting project for a client. The project/assignment was sourced by my uni's consulting society and is unpaid (team of 5 students). I listed this activity on my CV as part of my involvement with the society as a one-liner under 'Member of ... consulting society' which falls under the 'leadership and achievements' category.

I'm unsure how to/whether I should list this as a work experience instead and dedicate 2-3 lines to explain what I'm doing/have accomplished.

My concern is that
1) I'm not sure if I should state the client's name both because the owner is seeking to expand across the UK and has asked us to not tell anyone about the project.
2) It's not exactly proper consulting 'decorum' to reveal the name of your client on your CV or otherwise (even if they are a relatively young start-up).
3) It's part-time and unpaid and somewhat challenging to demonstrate actual results and expand on the job scope if I list it as work experience.

What do you think and how do you think banks and consulting firms will view my mentioning it as work experience (should I put it as work exp when applying to banks/asset management firms and not consulting firms)?
We are being mentored by an associate/consultant at a tier-2 consulting firm (think Deloitte, PwC etc)- should I bring this up on my CV or interview or not at all?
And any particular tips to format all this info as work experience?

Thanks! :)