Can I list Synchrony Bank as a source of interest income

If your bank is not on the drop down list of financial institutions from which you can import data, then click on "I'll type it in myself" and then enter your information by hand.

You can list any bank that paid you interest.  All you have to do is type them in as the payer when you enter your 1099-INT. Where does it say Synchrony Bank is not a source of income?  Do you mean you cannot import from them?  If so, that does not affect anything.  Simply type in their name as your payer.