Yes there are some other lenders that do non-recourse loans however they are usually shorter term bridge loans but still have a $500k-$1mm minimum.

To lend to an LLC you can find a fix/flip loan (short term, bridge loans) generally from a commercial lender or hard money lender. 

.- Bridge lenders - for value-add plays, or borrowers that don't qualify for bank debt.

I’m specifically looking for agency debt that bridges the gap.

You could try to find a hard money lender and if the deal is sweet enough they might give you like a bridge loan for a year or so while u save money to refinance into longer term debt

Then there is hard money used for the purchase or bridge loan to buy the property and then it will be refinanced out of the hard money into a Fannie Mae or portfolio/Non-QM loan.

It's certainly hard enough for those of us that "know" what we are doing but Bigger Pockets certainly does help bridge the gap for us.  

Currently, my team and I are looking at a 36 unit in Winston Salem (agency debt needed), a 33 unit LIHTC property in Charlotte (bridge loan needed), and a 24 unit in Concord (bridge loan needed).

I can get a 90% bridge loan plus 100% rehab for 10% and 2pts.

@Vante JayIf a complex required a large amount of renovation (an unstabilized property); the loan you get would not be a typical Fannie/Freddie loan but most likely a bridge loan.