Dead battery every few weeks... even after replacement

We have a 2013 Prius Plug-In, and it's run fine even after leaving it untouched for a few months at a time. However, in the last two months, it's stopped being reliable. It's parked in our covered garage, but it just doesn't come on every so often. First, it was a one-off, and a couple of hours later it turned on just fine. The next time, nothing helped – we had to get a towing car because jumpstarting wouldn't even work. He drove it to the garage and they replaced the 12v battery, claiming that this was the problem. Now it's dead again... despite the new battery. None of the interior lights were on, and only the daylight running lights outside.
We looked in the trunk and my husband noticed what looks like some kind of animal droppings – maybe that's a hint to what's wrong? However, I should note that while the engine didn't turn on, I did get all sorts of dashboard indicators... sometimes all of them, sometimes just the check engine light, and when I tried connecting it to the charger I got a big orange triangle with the exclamation mark to flash furiously. The charging indicator next to the plug never lit up however.

The problem is that in the city that we currently live, they don't even carry the Plug-In, so I'm not sure how familiar they are with this model. So before I get it towed to the garage again, it would be really helpful if one of you could give us some suggestions of what we should be looking for to locate the issue. Can it really be that a new 12V battery fails within the first month?

 

My husband just pointed out that they didn't actually replace the battery. On the bill, they wrote "found dome light left on and battery dead, charged battery to resolve. Tested ok". He also mentioned that for all we know, the droppings could be years old. We have parked the Prius in a closed garage for the last 3 years, but before (when we still lived in California) it was parked outside. I'm not expert enough about animal droppings to date them, and we rarely open the hood to be honest...