Your landlord's threat is likely unenforceable, but you need to move fast on documentation. The real issue is that "trained and certified" can mean different things, and your landlord might be banking on you not knowing the legal difference. A service dog trained for anxiety needs to actually perform specific tasks - not just provide comfort by existing. If your dog meets that standard, federal fair housing law covers you regardless of what the lease says.
I had a neighbor deal with something similar a few years back. Her landlord tried the same angle after she got a dog for PTSD, and the standoff lasted months because she didn't have clear written documentation from a licensed mental health professional stating she needed the accommodation. Once she got that letter - basically a statement from her therapist saying the dog performs necessary tasks - the landlord backed off immediately. The key isn't a fancy certificate from some online registry (those are often worthless); it's having a healthcare provider on record saying you need this animal and describing what it does for you.
Document everything your landlord says about eviction in writing if possible, and get a letter from your therapist or doctor stating you have a disability and need the service dog to mitigate it. Send your landlord a formal written request for reasonable accommodation under the Fair Housing Act. Keep copies of everything. If they continue threatening, California has agencies that handle this, and frankly, an eviction threat over a documented service dog is a legal liability for them. Don't wait around hoping it blows over - formalize it now.