12v Battery issues

I think you are OK with the bigger battery, as long as you make it fit in the rear compartment (you can buy the proper bracket). This battery is partially drained if it's a new battery. You should actually return/exchange it as it should have been 12.5 volts or higher when you first get it. That however is not the problem to your acceleration and fan coming on.

You should get the DTC codes read on the car. Most likely issue is the bigger HV battery going bad. That can mean big bucks $2500- $3000 new from Dealer.

 

Leading up to the failure, your car State of charge meter should have fluctuated from full to empty quite quickly. The warning lights coming on, passenger compartment fan coming on, sluggish acceleration are all symptoms of a failing HV battery. When you replaced the battery, you were able to reset the car for 48 hours before the ECU picked up the failing battery again.

This could also point to a failed inverter pump which overheated the car. However since you were to go 48 hours without any issues, this would point more towards the battery.

 

Could the problem/errors be because it is a larger battery? No. However measure the voltage on the 12V battery when the car is IG-OFF and when the car is READY. If the IG-OFF voltage is less than 12.6V, fully-charge the battery now. If the READY voltage is much less than 13.8V, the root cause of the problem is the DC/DC converter is not working.
Any ideas on where to find the smaller battery replacement? Your Toyota dealer's parts counter. That is not necessary if you install the correct longer battery bracket used with the Smart-equipped Prius. You don't want the battery to be loose in the back of the car as that is not good for the battery's service life and also constitutes a potential safety hazard.
Do bad 12v batteries cause the sluggish acceleration or is that another problem? Another problem
How do I check the battery fan for issues? Listen at the vent to the right of the rear seat back cushion, but that probably is not the issue in the dead of winter assuming you live in North America.

If you are serious about DIY troubleshooting your car's problems, get a digital multimeter and Mini VCI so that you can gain some clues about what is wrong.

 

no, but be aware that a fix like this is temporary at best.

 

battery balancing is serious business, involving load testing and special chargers. there are many threads here about how it should be done, and many threads from people who have just popped in another cell. some are disappointed, and some are comfortable pulling the battery every few months to rinse and repeat.

 

Those numbers aren't of any use outside of the LAN, i.e. the MFD, audio stuff, Nav system, which is not part of the OBDII bus. Makers know that self diagnosing cars are not as profitable as ones that just light up scary lights.