> all the banks would happily loan me $215,000 with $35k down, but no bank would loan me $15,000 to buy a $50k house.
This seems like a made up problem. Those same banks would also gladly loan you $40k with $10k down, and you could have the remainder to fix the house up or make huge payments every month. Furthermore, if you already have $35k it shouldn't be too difficult to save another $15k and buy the house cash. If it is, you probably can't qualify for the mortgage payment on either property based on your income.
This seems like a made up problem. Those same banks would also gladly loan you $40k with $10k down, and you could have the remainder to fix the house up or make huge payments every month. Furthermore, if you already have $35k it shouldn't be too difficult to save another $15k and buy the house cash. If it is, you probably can't qualify for the mortgage payment on either property based on your income.