My name is Marc. I'm a licensed attorney and I will be happy to assist you today. It's awful that your residence is bullying you and denying your service dog access to the dining room.
California law allows persons with disabilities to bring trained service dogs and psychiatric service dogs, but not emotional support animals, to all public places. Several different California laws set out the rights of people with disabilities who use animals to assist them. These laws include the Unruh Civil Rights Act, the California Disabled Persons Act (CDPA), and the Fair Employment and Housing Act (FEHA). (Federal disability rights laws, such as the Americans with Disabilities Act (ADA), also protect the rights of people who use service dogs and emotional support animals.
A "service dog," under California law, is a dog trained to help a specific individual with a disability with services such as fetching dropped items, minimal protection work, rescue work, or pulling a wheelchair. Second, it is further limited to dogs that are trained to help individuals with their specific requirements.
I'm not sure what kind of service you dog is being trained to provide. If the dog is meant to provide only emotional support, you may not be protected by the law. But note that California doesn't have a separate definition for "psychiatric service dog," but a dog that is individually trained to help a person with a mental disability with specific requirements is considered a service dog, and an individual that uses such a dog is entitled to the same rights under the law as someone with a physical disability that uses a service dog.
Note that your residence cannot require you to "prove" that your dog is a service dog. A service dog is not required to be registered, certified, or identified as a service dog. You mentioned that your dog is in training. Please also note that even dogs in training, along with their trainers or owner-trainers, are also protected by these laws.
Do not be bullied! If you feel the residence is mistreating you by illegally discriminating against your dog or in other ways, you could seek help through the CA Dept. of Social Services - Adult Protective Services: see http://www.cdss.ca.gov/agedblinddisabled/PG1298.htm Here is another helpful service: http://canhr.org/factsheets/nh_fs/html/fs_preventabuse.htm
I hope you resolve this peacefully and soon! Also, I hope I've provided you with useful information so you can better understand your issues and options. If so, kindly take a moment to rate my answer, since that is the only way I can receive credit.
Good luck and good health!