With me, i applied with four banks in parallel and two banks declined my application (after 8 weeks reviewing it: Chase and Citizen Bank) and two banks still reviewing.

Try US bank.

We have a sizable amount of cash for a down payment, but does this need to be somehow tied to the LLC, other than in our personal bank accounts, prior to going apply for a mortgage?

Remember, the bank is a partner in this investment and they are getting interest paid to them on whatever part they contribute (loan) to the deal.

Since the COOP building has an underlying mortgage, if one or several such wholesalers can't pay their monthly fees, the COOP cannot pay the underlying mortgage, the bank can foreclose on the entire building.Condos does not run into this same problem, has it has no underlying mortgage, though banks have foreclosed on sponsors.

Have you tried asking your HELOC bank to increase your credit limit on it?

My local banks do not have knowledge in this area, and perhaps that makes sense as the local banks often don’t deal in this area as it is a federal loan.

But since I owned multiple units that were collecting income I could go through the eviction process without any repercussions of my personal bank account.  

Not my cup of tea, but Boston has a lot of great things going for it that in the long run I think you could make it work.If by multi-family though you mean a 5+ building that you're presumably going to look for in the Midwest or South...it's in its own bubble as a disturbingly large number of people who made bank in the last 10 years of the up market are shifting their focus to a larger scale asset and are purchasing these distressed properties which is driving cap rates down because people need a place just to park their cash. 

I could get a travelers cheque or a bank draft.