One other thought .........
Have you actually measured the voltage when it appears to be discharged?
Here's one consideration - this thought was sparked (no pun intended!) by knowledge of a common problem and aided by your comment about the charging voltage being 13-something:
Generally, the charging system will give over 14V on a Rocket - now that is with a reasonably charged battery of course - if battery is discharged, then even when running, the voltage will be low (that is no different that how your battery charger displays the voltage - as it becomes more charged, the voltage of course goes up until it is saturated);
yours being 13 could be just a product of the voltage having to recover from a discharged state;
however it could also be a sign of another issue, quite common, especially on Touring I have noted;
the Ground lead - the smaller gauge one that comes off the MAIN negative goes to the Bike's main grounding scheme AND to the R/R negative goes via a single-terminal connector about 9" or so from the main terminal - that connector frequently burns and you have a high resistance connection there.
Now if that is the case, then a) you will not be charging adequately and b) you will have a voltage drop across it
In the 'b' case, your battery itself may not be as fully discharged as appears by its behaviour, but because is below its previous threshold for operation, may discharge enough for it to render it as if it were.
Highly speculative but simple enough to check - look for that connector and inspect it for signs of burning.
(
@Kevin frazier can attest to that one too

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