She proposes to use her security deposit money to cover another month's rent, to be paid once it's released.

Feature/Service BlueVine Axos Bank Mercury Novo Interest Earn interest on checking account balances High-yield savings accounts No interest on checking; conditions on savings No interest on checking Fees No monthly fees, minimums, or incoming wires Low fees, but some services might have fees No monthly fees No monthly fees, but specific activities may cost Account Management Free incoming wires; easy online management Comprehensive online banking, specialized for real estate Easy to open multiple accounts; digital management tools Easy integration with tools; user-friendly interface Specialization General business banking Tailored for real estate and property management General, with a focus on startups and tech companies General small business banking Accessibility Limited physical branch access Online-focused, limited physical branches Fully online, no physical branches Fully online, no physical branches Integration Basic digital banking tools Some integration capabilities Strong API for custom integrations Good integration with business tools Cash Handling Not ideal for cash-heavy businesses Can handle cash through ATMs/restricted branches No cash deposits No cash deposits

So you would ask for, i.e. 45 day feasibility period (the time in which your deposit is still refundable) and then if by day 45 you feel confident and can start to move towards a close, then you will ask for a certain amount of time to close, but during that second timeframe, the deposit is now NON refundable.

You will increase the rent and security deposit because of the dog's liability.  

Do you know if your security deposit increase is legal in your market?

If speed is of the essence you could offer 100% security deposit return if they’re out 30 days faster than required.

Someone I know owns section 8 properties, he doesn’t have any issues and the rent gets deposited directly to his account.

First month's rent and deposit have been paid. 

That means effective showings, effective applications, checking references, background checks, collecting cash or cashier checks for first & deposit and waiting for them to clear, starting the eviction process immediately if they are a no pay, etc.  

You don't actually hold the deposit, from my understanding.