My tenant violated a "no pets" policy. We gave them a 3-day notice. What if they want to bargain? (Increase rent for month?)

The notice period in the lease protects the tenant to the extent that it give it a minimum time frame to remedy the alleged default in the notice without being able to be evicted. Once the notice period elapses, the tenant may be in default under the lease and may be subject to eviction (depending on the default/remedy provisions in the Lease).

As a landlord, you may elect to not evict the tenant for not curing the default within the notice period and may give the tenant as much time as you deem appropriate (a business decision and typically based on the relationship with the tenant).

Any rent or security deposit increase should be governed by the terms of the lease, and if said terms are valid and enforceable. You may negotiate with the tenant and amend the lease for increases in rent and security deposit in exchange for you allowing additional time or for the tenant to keep the pet.

It is very important that you comply with the terms of the specific lease. My comments are based on general principles and may not comply to the written agreement with your tenant.

I agree with my colleague. If the original lease does not call for extra fees in the case of breach, you may not be able to charge extra. If you want to negotiate a new lease and charge them a pet deposit and/or pet rent, you should speak with a local landlord-tenant attorney as to the maximum amount allowed.

First of all, was this "no pets" policy in effect at the beginning of the tenancy? You should be aware that it is illegal in Los Angeles to allow pets, and then disallow pets, for the purpose of terminating a tenancy. You probably cannot charge more rent if the unit is rent-controlled. If you want to give the tenants more time to re-house the dogs, that's up to you.