How can I avoid paying both a pet rent and a pet deposit, here in California?

Choose a different place to live. If the landlord requires a surcharge and deposit to allow a pet, that is up to them.

Confirming it is legal to charge pet rent and pet deposit in addition to regular deposit?

You pay what the landlord wants or you find another place to rent. You hunt around to find what you can find the best deal on.

Don't call it pet rent. Make it a pet security deposit, in advance, or you can get into denials that there's a pet or pets there. You could make it high and say that if you find no evidence of damage, stains, smells, and complaints from neighbors, you will apply some of it to the next year or refund it. That will encourage good behavior.

Funny, I answered this on your other post as though you are the landlord.
Landlords have seen the worst when it comes to pets. Most won't rent to them at all, so be glad for what you get, and keep them from using the apartment as a bathroom. That is usually the most frequent complaint and it's extremely expensive to clean.
Offer to pay what the LL wants with an invite over in 3 months. Show them how your animals have been practically invisible and ask if he will consider a reduction.

Threads merged please use the Answer options below if you have a follow-up information or question.

Please stop posting several posts on the same issue.

I have merged them, if you want to add more info just answer your own question.

A landlord does not have to rent to anyone with pets.

They can limit the type of pets, They can limit inside or outside pet. They can limit the number of pets, even the breed of pets.

They may charge more if there is a pet in the home, and they may charge a pet deposit, since most pets cause additional damages.

How does the new carpet play into your animal issues?

I do know most landlords will have to ( perhaps not required) to professionally clean a carpet after there are pets in it, and some renters would never move in to a house where pets were on the carpet,