Seeking a play-based Mandarin tutor for a 6 year old

You can probably find a UC Berkeley student who likes kids-- Cal likely has a job posting function, and you could also go over to the Education and Psychology departments to see if they have bulletin boards where you can post a notice. I did this with my kids (when we lived in LA) and we found two very nice students to come over and do arts, crafts and games-- Mandarin immersion.

Hanwen school may be an option too. 

We have a similar family background and this sounds like a great idea! I hope you are able to find a tutor that works. FWIW, here are some other Mandarin options that my family has or will soon try out:

• Shu Ren International School - Mandarin immersion preschool and elementary school that also offers summer camp. We have had 2 preschoolers here over the last 4 years. Good balance of play and instruction.
• Hanwen after-school care and summer camp - the after-school care Mandarin classes are fantastic and my daughter (also 6) has made a ton of progress in writing Zhuyin. However, they seem very structured so maybe not a good fit if you are looking for more of a play-based vibe. After-school care has offerings for non-Mandarin speakers but summer camp requires fluency.
• Su Yun Chinese Learning Center - we have not tried this one yet but are signed up for summer camp. They offer a variety of classes and camps that look very activity-oriented. They also offer tutoring so might be a good place to contact for leads.

Also a note about playing with other kids - our experience has been that kids will speak Chinese with their instructor but English with other kids, so possibly something to consider before setting up a play group (though this probably depends heavily on the language backgrounds of the individual kids).

Good luck!