Yes, I have a Toshiba that I have cleaned up for a family member that does have an internal rechargeable battery for the CMOS and real time clock.
Before you start hunting for the battery though, it sounds more like a power supply circuit problem to me.
The question to answer, is if you remove the regular battery and unplug the computer does it loose the real time clock after a few hours?
If the real time clock looses a lot of time, or even resets without any other power to the laptop, then yes, the internal battery needs to be replaced.
However, if it keeps time, then you have an internal power problem inside the laptop. Most likely is that one or more battery contacts are not making proper contact between the battery and the laptop.
As the charging circuits and power circuits are separate, it is possible to charge a battery but not be able to run off of a fully charged battery.
You will need to inspect both the battery and the contacts inside the battery compartment to make sure that none are bent, broken, corroded or missing.
Chas