I don't have an RC51 yet. I have a YZF600R, but I hope to have an RC soon, and as such am continuing to read up on these motorcycles to best prepare myself for when the time comes for purchase.
That being said, I wanted to ask a couple questions about how you got your readings. When you say that you got 13.5v or so at the stator, how did you do that? Was that at the stator itself, coming off the stator? The reason that I ask this is because generally, stators put out AC current, and a lot of it given some revs. The voltage can vary pretty widely. While I don't know what the voltage is for an RC, for my YZF, it can be 20 to 50 volts depending on what speed the engine is turning. It is the rectifier / regulator's job to "throttle back" that voltage to a more reasonable 14 (or thereabouts) volts, which should charge the battery as you run about town. The regulator's other job is to change the voltage from AC to DC, which is what the bike's electrics run off of.
What I propose is a test - run the bike around town, get it quite hot, and then measure the voltage at the battery terminals. Voltage to the starter is going to be significantly higher than the voltage to the digital dash. If the bike is hot, my guess is that the R/R is also not able to disapate heat as it should and could be failing. Though, as I think about this, having a new OEM unit fail within the first week would be unlikely. My other thought is that the starter may be going out. If you're getting proper voltage back to the battery once the bike coolant temp reads 230 or so, then I would say that your new R/R is probably okay, and you might be looking at a starter rebuild.
For peace of mind, you can pull the stator cover off and make sure that you don't have burned coils or burned wires leading out of the stator. Pull the connectors (if you have any) to see if there are any burn marks there. I highly doubt that a Honda would, but single phase Ducatis are pretty famous for having burned connectors and melting plugs. Worth a check, at least, just for peace of mind. Again, check the voltage back to the battery, to make sure you're getting the proper voltage back to the battery as per the service manual. I have an RC service manual that I can get to, but I am on vacation currently. Once home, I can look it up for you. I won't be home until Monday.
Check the plugs going to the R/R. They should also be clean (i.e. not burnt).
Finally and JUST as importantly, check the ground wires. A faulty ground wire will cause the bike not to start, or be hard to start. Start with the ground wire from the battery. Having a ground that does not fully seat can be terribly frustrating.
Feel free to add some dielectric grease to the connectors to keep the connectors free from corrosion from water (if you ride in all weather conditions, as I sometimes do). A pinch of preventative maintenance is worth a pound of cure...
Anyway, as I am typing this, I am thinking that you might have a faulty ground wire.
Hope this helps.