Smart power supplies for turntable

Whether it works better with battery all comes down to implementation.  The CA with active feedback is going to be more immune to supply variance. 

Electric bicycles are typically 24-48V not 12. The nominal 12V lithium vehicle batteries are as high as 14.8 fully charged draining down to 10. A 12.6V LiPo for RC may be more appropriate. You could test it with an 8x AA battery holder and some dollar store batteries.

Their battery supply lists the output as 12V, but it also says NiMh 12V, 5000mAH. As a guess only, it is a 10x series NiMh, 12V nominal, but to as high as 13.2, goes down to 10.  This then supplies the turntable that has the DC motor controller with servo speed feedback back on optical sensor.

The output of the fixed supply is 12V. This was my point with the wide range of just using a battery. Odds are it will work, properly, if reasonably designed to a volt or two below or above 12V, but the only one who can give a correct answer is ClearAudio.

To clarify, I have a Clearaudio turntable with a 12v DC motor. Clearaudio indicates  that their DC power supply is voltage stabilized, yet they sell a separate 12v battery power supply. This suggests the stock power supply has some limitations. 

To me it suggests they can sell it and make money. Expectation of performance is almost as good of a consumer motivator as validated performance.

The operating range on the ClearAudio AC supply makes it sound like a switcher?  That is not inherently bad. It will mean it will handle your 113V or pretty much anything you throw at it without too much issue. Depending on how it is designed, the output voltage may be totally independent of the input, or slight variation.  The turntable also has active speed control based on a speed sensor, so that is ultimately setting the speed, and will also adjust up and down as needed.  I don't think you would see any variation in turntable speed if you tried to measure it. Not even a tiny bit. You would need a wow and flutter meter. Realistically, the bearing are causing more speed variation, not to mention non perfect placement of the record hole, etc. etc.

I would be far more concerned about just general "noise" which you seem to have addressed at some level. Maybe you can borrow a power regenerator and see if it meets your needs.

How about just go with Cin. That would be more accurate.