So can having a weak 12v battery affect mileage?

 

In theory yes but not much. ~(0.2) mpg ish.

Now some theory. The vehicle loads are mostly "passive" or behave that way. So OHMs law applies. So volts squared divided by R is power. That power comes from fuel mostly. Any time volts is higher than necessary, you are wasting power/fuel.

Regarding your vehicle. The DIC will show you the 12V system voltage. Most newer vehicles have fuel economy inspired 12V system management. Yours is one of them. If your DIC voltage is getting down into the 13V range, the system most likely thinks the battery is charged and good. It will take about 5 to 30 minutes to get down there. If not the DIC will almost never drop below 15V. Remember standard integer rounding is used. 14.5 displays as 15, 12.8 as 13. and so on. So a 15V on DIC for more than an hour driving in conditions described below indicates possible problems.


NOTE: the use of high blower in air conditioning, windshield wipers, headlights will all cause the 12V system to boost to around 14V.

Now the bottom line. The DIC at 15V all the time and OHMs law will conspire to steal about 0.2 mpg maximum versus "fuel economy" mode allowing the voltage to reduce into the 12.8-13.4 range. 0.2 mpg may not be a big deal to the individual but CAFE for OEMs is a big deal.

 

I actually thought it might be more since the Hybrid battery is always charging the 12v one.
But 0.2 is pretty small. Good info, I checked out the DIC before and after changing the battery. And it followed your logic.
Thanks.