In my state, calling something a deposit (pet deposit) is not the best option because the property owner needs to document pet damage to retain the pet deposit. 

Hi Brooke,my advice to you is if you already haven't started using the rental property calculator under the tool tab definitely start. 20k for a down payment is probably a fair number from the bank as you'll have your initial down payment, closing costs and deposit. 

mortgage with a large deposit from a 1031.

Make an offer with a refundable deposit of $5,000 subject to a 10 day inspection period so they know you’re serious.

If tenant is willing to pay 1 month pre-paid rent  ( $750) in additional to the first month's rent ($750)  and security deposit at move in, then this pre-paid rent can be spread out over 11 months to lower their monthly obligation from $750/month to $681.81 so they can meet or almost meet the rental qualification of income being 4X the rent.  

The key is to find a bank that is aggressively seeking loans and deposit accounts.

For example, I often see Buyers purchase property thinking there is a $1,000 deposit but then the tenant claims it was actually $2,000 because they paid the last month's rent.

At that point you will have the (2) months bank statements needed that do not show a "large deposit" that would need to be sourced and seasoned.You might also want to look at other banks as there are more programs becoming available for 2023 up to 100% financing. 

That includes security deposit. 

Not soliciting just want some advice on a high level.PP $348,000FHA Loan 3.5% DownInterest Rate 3.125%Closing Cost Estimate $8700PITI $2650Property 4 unit Built 1974Sqft 2978Unit Mix 1 1bd1ba rent -$725 3 2bd/1ba - $780Gross Income $3065Deposits held (refundable 1st months rent)Fully occupied Tenant pays water/gas/electricity (individually metered) Upside Increase rents to market about 10-12% under market Implement surety bond for non refundable deposit Add on site laundry for additional income.