the mediator will recommend to the judge that my X and me share…

That's not exactly true. The court is required to do whatever is in the child's best interest per Ca Fam §§ 3011, 3020, 3040, 3041.

In determining what is in the child's best interest, the court will consider whether there is abuse of the child, substance abuse of a parent and criminal history of parents. But, the court also considers many other factors such as the child's preference, home environments of both parents, family network of both parents, being separated from other siblings, etc.

Thus, there are other factors to consider. Here's a good article that outlines all of these issues:

http://www.kinseylaw.com/clientserv2/famlawservices/childcustody/childcustody.html

There are legal aid offices and pro bono attorneys throughout the state that may be able to assist you. I would recommend that you contact the California State Bar Assn. (http://www.calbar.ca.gov/) and ask for help locating an attorney in your area.

When the mediator gives the report, you should be allowed to question her about the report, considerations made, whether or not she consulted what the child wanted, etc. Also, you can argue your position to the court and make your case for the court not to follow the mediator's recommendation.

I will say that this will be difficult because a mediator is supposed to be impartial and be acting in good faith. Thus, you'll have to be able to convince the judge tht the suggested arrangment will be detrimental to the well-being of the child and that it is not in the child's best interest because she's always lived with you and such a change could cause mental problems (anxiety, depression, etc.) for your child.

Again, if you can get help from a local attorney putting this together, it would certainly be easier for you to put your arguments into a good legal framework.