Is this normal battery behavior?

This is fairly normal. The ac uses a lot of energy, and when you pull away from sitting after a while the hybrid system relies on driving power from the electric motor while the gas engine is running at idle and warming up to operating temperature.

Mind you it does get more noticeable with a older used battery.

 

Sounds like the new battery is not that strong. Also sounds like your not doing anything to mitigate cabin heat. You should be using a weather tech windshield shade that is form fitted and insulated and hope you have a good tint on the windows and I keep the package cover extended in the back cargo area and have a form fitted piece of foam insulation covered in foil. That sits on that rack so no direct sun hits the back battery area. Really helps keep the cabin cool. The sun is so strong now and your letting the cabin get super heated and that kills the battery. Probably attributed to the original battery failing. Shade available at the Priusshop.

I recommend u purchase a hybrid battery charger from hybridautomotive.
Occasional charging helps and you will see the difference between a good freshly charged battery and the present battery's condition.

 

... and regardless of what the so called "reconditioning" process your installer used, it is still an old battery of recycled old components. You cannot make new from old.

To answer your 2 questions,

1. Yes, the Prius draws more heavily on the HV battery when it is first warming up, but not especially because of the heat. An older pack will be weaker than a new pack. You may benefit by letting the car get through the first stage of warm up, which takes about 55 sec from when the ICE starts (with the A/C off) and is signalled by the ICE shutting down. Wait until this happens then start the A/C before moving off. The only way to rejuvenate your car is to install a truly NEW battery and that means a new battery from Toyota.
   
2. No, but the battery does not like to be hot, so depending on the battery's internal temperature, the system might ease off the load until it is cooler. If it does that you may see poorer performance. I'd make sure you cool the car as quickly as possible to 73-74ºF. Open the doors/windows to let excess heat escape and then get the A/c going. As a preventative measure, use a window shade, if you have one, for when the car will be parked in the sun.

hope this helps

 

The behavior you're talking about probably doesn't mean anything is wrong with the battery except that it's older. And I think dolj meant that buying a refurbished or reconditioned battery means it's not going to last as long, not necessarily that it's going to stop working in the next couple weeks. If the battery is in risk of immediate failure, you would probably see the charge level bouncing up and down every few minutes.

 

Exactly.

No, I am not arguing against reconditioning, per se, but rather the proliferation of battery "rebuilders" doing who knows what and passing off the rebuilt battery to the end user who has the expectation that it will be as good as new. The stories we see here repeatedly is the that these batteries generally are not lasting 6 months, let alone 3 years.

Your statement here is exactly is what I'm talking about above in regards to user expectation.

In essence, there is a place for reconditioning (as understood using the HA charger/discharger regime), as a user/owner self service, but to be useful it has to become an ongoing maintenance. I personally believe that reconditioning will not make a battery last forever, but a battery reconditioned in this fashion will last a lot longer than one not having the benefit of this process. Notwithstanding this, I still stand by my statement an old battery is an old battery and will never be a new battery.