Since you're using CT-657, I suppose that's a C corp return you're filing?  Either way, ProConnect Tax nor Lacerte support IT-657 and CT-657.

If you simply attach that as PDF, you risk the form not being processed.

Can't understand why Intuit Support is often not familiar with the very programs they support and would have told you to simply attach it for e-filing.  The e-file attachment screen has a big disclaimer that says this much:

NOTE: Do not attach any document that needs IRS approval or attention.

Yes, NY does mandate e-filing.  You are, however, not required to e-file returns that are not supported by your approved e-file software or those that can't be e-filed on the NY Department of Taxation and Finance's website.  See this webpage for more info:

https://www.tax.ny.gov/tp/efile/mandate_participants.htm