Deposit increase?

Tenancy Deposits

Based on my rental charges I am able to request %C2£700 deposit for one of my properties.

I have a tenant who has been in a property for over a year. Based on the original interview (never having kids, never wants a pet, no smoker), I agreed to a lower deposit of %C2£500.

The tenant has now requested a pet. To allay my concerns re the low deposit, the tenant has offered to increase the deposit by %C2£200 (still under the maximum deposit that I could ask for for that property based on the rent).

The Tenancy Deposit Scheme has indicated that to change the deposit amount I would need to a) generate an Addendum to the Tenancy Agreement outlining the agreed increase to the deposit, and b) following that signed Addendum I would need to then end the existing protected deposit, and create an entirely new one for %C2£700.

My questions are

a) Have I understood the above process correctly? and,

b) Am I right in thinking that I am unable to charge the tenant for protecting the new deposit amount, even though I've already paid for the original deposit protection and I am only doing this as they have offered to increase the deposit?

The tenant has informed me that the puppy has arrived and is in the property.

Thank you