The 4/10 loan limit is not "by law". It is each lender's decision, but most lenders are selling their loans to Fannie Mae and Freddie Mac, so they have to meet Fannie/Freddie guidelines, which limit the number of residential loans a person can have.
So what you're looking for are:
1 - Commercial loans on residential property. "Commercial" in this case doesn't refer to commercial property, it refers to the fact that they are lending to a business rather than an individual, or that it is simply the commercial lending division of the bank. So your LLC would typically be the buyer and borrower, but the property can be residential. You just need to get out there and talk to some local lenders and see who offers these types of loans on residential property.
2 - Portfolio loans. These are the same types of loans that you have been getting, but if the lender is going to keep the loan in their portfolio rather than selling to the secondary market, then they don't have to meet Fannie/Freddie guidelines, so they can give you as many loans as they feel comfortable with. This will be from a local bank, not Bank of America or Chase.