If that country is part of the Hague Convention, a local official can handle the signing and attach an Apostille authenticating the document(s).

I would focus on improving your credit and obtaining a more stable income so that conventional financing is an option. 

If not, I'm guessing it would be treated as a traditional rental loan, either conventional or DSCR, using market rents rather than the rent in the lease agreement.

I can put 20% down; the rest could be HELOC or a conventional loan. 

This is what I am encountering, in the current Mortgage Industry.If you think you will go FHA, Conventional, FHA 203k, etc. and then Quit Claim the property, to a LLC, or a Land Trust you run the risk of the lender discovering a Title Transfer occurred and activating the "Acceleration Clause" or "Due on Sale Clause" that requires the loan to be paid in full, within 'x' number of days.

I can make the down payments in the smaller market on cash-flowing deals, but these bigger markets require 40-70k to make a deal the conventional way, and I just don't have that to invest.

I would recommend reaching out to @Denver McClure as he may be able to help with a non-conventional loan

Since you have work history, if you get a job with salary income and a credit score above 640 (680 realistically), a conventional cashout refi is within reach as soon as you reach those metrics.

I know of a lender around 7.3%, but curious what everyone else is seeing.Keep in mind, we're talking DSCR, not conventional.

Higher loan balances we would likely see an opportunity in the future to refinance again due to lower rates so consider a 2-3 year prepayment penalty.Of course if you qualify for Conventional Financing (Fannie/Freddie), there are no prepayment penalties so just calculating out the return of investment timeline on point buy downs.Good luck shopping and let me know if you want to talk further!