Your calculations are correct, bank will loan up to 70 LTV less first mortgage balance.

Check with you local bank to see if they can help.

Typical bank draw cycles are at least 30 days plus a few days for the bank to process.If your contract stated payment immediately upon completion then you have a solid argument.

In March 2016 I had previously attempted to finance a property through a big bank.

Please tell me what it is and I'll explain how this works.

Should I use a bank or what type of source for this kind of financing?

Now, In my first offer I included an inspection contingency and also it is all cash offer, the bank countered for Highest and Best, due to multiple offers.

Talk to some local banks about portfolio lending.  

That is how appraisers do it, that is how banks do it, that is how you should do it; not by listening to out of state bozos that know nothing about the market and try to value residential real estate via price to rent ratio or CAP rates that have absolutely nothing to do with the value.

Explain that you were able to give a "pass" this past month, but rent needs to be on time in September and every month going forward.