The issue with the 15 VDC is where the RR gets the sense wire signal from. Check the voltage reading on the sense wire. If there is a voltage drop from the battery of say 1 VDC, the RR gets this voltage signal and compares it to the RR reference voltage of approximately 14.2 VDC and adjusts the output to 15.2 VDC, just math - 14.2 + 1 = 15.2 VDC. The unfortunate issue is that the voltage drop on the sense wire, if there is and probably will be, will probably be constant so the RR will continuously put out a greater than RR reference voltage. Your output voltage would indicate that there is a possible 0.8 VDC voltage drop on the sense wire.
You could consider a relocation of where the sense wire is connected into the electrical system to get a more accurate voltage reading without going to the battery. This schematic:
has been modified to show how this can be done. Change at the starter solenoid is not required. In this schematic the sense wire is monitoring the RR output for any fluctuation in electrical system voltage. As the electrical system voltage decreases, indicating an increase in electrical system amperage, the RR will compensate to return the electrical system voltage to approximately 14.2 VDC. Conversely, if the electrical system voltage goes up, indicating a decrease in electrical system amperage, the RR will compensate to return the electrical system voltage to approximately 14.2 VDC.
This modification eliminates the issue of voltage drop across old and tired wiring, and it is easier on the RR and charging system.
As for the battery, after the engine is started, the battery accepts enough power to bring it back to a 100% state of charge and this is at a rate that is ever decreasing, after which it reduces the intake according to the internal resistance of the battery. The battery will take in only what it needs and at this point it results in the battery accepting a trickle charge of say 2-3 amps. Batteries are abused and fail prematurely when the battery is used as a load distribution centre, specifically if loads are attached to the positive battery terminal because you cannot direct where the amperage is going when there is more than 2-3 amps at the battery positive terminal.
The other issue is that the battery is a small load on the electrical system until the electrical system voltage drops to and/or below 12.6 VDC - battery voltage. This generally occurs at low engine RPM. As long as the electrical system voltage is greater than 12.6 VDC, the battery is a consumer not a provider.
Just my thoughts on your issue. Good luck.