In the quality B class neighborhoods I was invested in during the last downturn, rents stayed the same in 2009, but went up all other times, and I missed exactly zero rent payments; in the C-class properties, rents didn't go down, they stopped ...

Most of these properties are between 70-110k with payments around 400-500/month and rent out for about 1200 a month.

I don't need to make a profit, but I need to keep my payments as low as humanly possible, including property management fees.

What down payment should I offer?

If your mortgage payment is $5K a month and the income is $6K a month it will probably hurt you ability to get another loan. 

I'll add the shipping fee to the payment thru my paypal account so, you can pay the mover on my behalf.

The Chapter 13 plan calls for the arrearages to be repaid over 60 months, but does not include the original mortgage payment anywhere in the plan.  

If you're using a HELOC for a down payment, or for that matter payment in full, I would think most banks would allow it. 

Option 2 - Higher monthly management fee, but spread out payment of leaseup fee.   

Very little risk to the seller with that large of a down payment.