The 4-unit limit is I believe a Fannie Mae and Freddie Mac limitation: http://www.freddiemac.com/singlefamily/news/2017/1128_loan_limits.html
Anything beyond that, and your investment is no longer backed by the government, which is why you can get the lowest rates on conventional loans (i.e. less risk).
The question to ask yourself is, is a commercial loan all that bad? How much does it affect your cash flow?
Putting a higher down payment, at least 25%, will also se you up for better rates.
Private money and maybe checking rates at your local banks! Bring the current offer and see if they can meet you halfway on your proposed 6plex?! Doesn’t hurt to ask and sometimes in the process big banks and little banks offer up insight to each other’s inner workings! Good luck!