At this early stage and being across country, I just haven't built the relationships in order to pull that off yet.

That, combined with our approach to pricing, staging, and presentation, all have resulted in very high-quality tenants that exactly fit our target demographic.  

You can add late fees automatically and there are no excuses for things like, arranging time to meet, or not having another check...

I'm looking to owner-occupy a fourplex (not yet under contract) and would prefer to arrange for financing ahead of time.

If this is a non- prohibited arrangement and thus allowed, how long does it take to set up the self-direct fund and do I use an attorney, my investment company Merrill Lynch or some other professional?

You are absolutely within your rights to determine at this stage of the process to deny this applicant since there is no signed lease. 

Another thing to keep in mind is if you need to refi to get money out in a couple years in the future for your next down payment there is a big possibility you will be doing so at a higher interest rate.If you pay more down you are not reducing what you are paying monthly just shortening the number of payments in 25-30 years.It looks to me you are in a growth stage and having a stable predictable property with more debt (paying off at a regular rate).

So I'm in the rehab stage of a BRRRR and thought of a question regarding seasoning period. 

Are there any potential pitfalls or specific things I should do in regards to a lease with an arrangement like this?

How old are you, and what stage are you in your life?