We told them, that damn chain drive was shot, we told them that weeks before and then it jumped just like we said it would, hell no that's not on us!". 

Using a HELOC to pay for rehab.It will have some decent monthly cashflow, just not sure about some of the other numbers.  

We do have decent w2 income and 770-800 credit.

That could put you in 6-10 deals all over the U.S. and provide you with fairly decent passive income. 

It may be too small for a decent property management company.

Traditional methods offer me a pretty decent amount of financing to work with, but I’m wondering if less-traditional methods would allow me to do the same with less money down.

Prices usually run $600k - $900k in decent areas, for properties that don't need much work.With that said, if you plan on moving in, either solo or with the family, you could find one that needs some work and put in the sweat-equity. 

I have a decent chunk of money in savings (apart from an $8k emergency fund), no debt, no pets, no kids, no significant other, own my car and have very low living costs for my area (under $1k/mo for a room in a shared house, all included, off-street parking, have been here for 4 years).

Given this, I'm assuming you have a decent job.

I know it's a long shot and likely impossible, but if there's a way, what do you think would be the best bet?