I'm almost embarassed to post this, figuring I must have missed something. As far as I know, the registry keeps neither the date or the time; only settings on how they and other types of data are formatted.
This snippet will extract the date and time:
set dt=%date:~10,4%%date:~4,2%%date:~7,2%
set tm=%time:~0,2%%time:~3,2%%time:~6,2%
You can then use the %dt% and %tm% variables to rename your file:
ren fileA.xxx fileA-%dt%-%tm%.xxx
The above is just an example; you can format the renamed file label as you wish.