The shortest possible answer...the SBH-13 will function on any supply that can provide 12V DC and is rated at 500mA of output current. The NC-88B is a no go.
Longer answer...I've used three different 12V supplies on my SBH-13, each one has worked with no issues. The SBH-13 just wants 12V DC, at some current obviously not exceeding 500mA. The circuitry in the charging cradle does the rest.
I dug into the Yaesu AC adapter situation a bit and it looks to me like the PA-48B is discontinued, and the current equivalent is the SAD-25B - according the Yaesu website this is the AC adpater that is specified for use with the SBH-13. The FT-60R I purchased a year ago (Summer 2019) came with the SBH-13 cradle and an AC adapter that says "SAD-25" on it. I don't know why my adapter doesn't have the "B", it may just be part of Yaesu's part numbering..in any case, the specification is still 12V DC at 500mA.
Also...all of the above applies to the US version, which will not allow battery charging via the DC power jack on the radio. Since you mention in your question charging via the power jack, either you have a non-US version, or this was previously a feature on the US version FT-60.