The best way to avoid having to source where the funds came from is to deposit them 3 months prior to the application. 

She also said the other units on the block are cheaper and do not require a deposit.

Don't forget about the deposit.

If you are concerned about competing with cash offers put up a MUCH larger deposit and do not have a mortgage contingency.

It would be smart to create a separate account for all your deposits to make it easier to track the income.

We also slightly lowered the security deposit, knowing that coming up with almost $4K at the holidays is not the most appealing thought for tenants.

Are vacation PMs depositing their own money to maintain the minimum in client's trust accounts? 

At the time of move-out check if there is any difference and deduct from the security deposit.

Any deposits and pre-paid rents get them at closing or you will never see them.

The reason you want them to have renter's insurance (with you listed as additional interest and/or additional insured) is because you want to have someplace to go outside of your own insurance if they do serious damage in excess of what the deposit covers.