Inverter Coolant Pump issue?

Our 2003 Prius recently experienced some malfunctions which I think are related to the inverter coolant pump but I wanted to get a second+ opinion.

I've searched and read regarding this situation but I wanted to post our specific experience in hopes that it might help confirm what's happening.

About a week ago, the orange triangle popped up along the battery and check engine lights. I thought it was battery related so I reached out to a friend of mine who has had similar problems in the past. He recommended checking the voltage and then replacing the battery if necessary. I did some research regarding 12v batteries on here and even posted about it (despite an already littered field of questions relating to 12v batteries).

On the original battery in the equation here, the car was functioning between 12.7V - 13.7V in diagnostic mode. I'm okay with that based on what I've read here. I noticed that the A/C running on high seems to pull on the battery some and considered this part of the equation.

I left the A/C on and after about 5 minutes the voltage dropped from the somewhere 13V+ to somewhere in the 10V range. Left it going for a few minutes more to see what would happen. It eventually dropped into the 8V range after a couple of minutes. Cue flickering display/restart. I decided to turn off the A/C. Gained about 0.1V - 0.3V back but never really went above that. I put in park to run a few minutes, and then left the car. The following day I tried this again with the same result. I figured the battery may have been bad and was just using built-up charge. Also considered something with the charging/electrical system. Figured the battery would be the first step since it's all connected.

I went ahead and went with a Toyota battery. Spouse felt more comfortable that way even thought the cost was considerably more. Installed it myself. Left the parking lot of the dealership, and two miles down the road we have orange triangle with battery/check engine lights on. Called the dealership and went back by. They checked the NEW battery (which was showing around 12.7V - 13V+ and they said the electrical system was fine. Insisted that there were no codes present that they checked because that costs an extra $79.95 and I didn't authorize that. They did, however, present me with a list of other random things (not at all related that they feel the car needs). With this in mind, however, I was rushed to the cashier to close out and the service guy ran away quickly (trying to get rid of me). I did get the trouble codes they said they didn't check from the sheet the cashier left on the counter when she went to make a copy of the tech's recommendations upon request.

Trouble codes are the P3125 and P3130 -- the familiar Inverter Coolant Pump and Inverter Assembly codes. A friend also confirmed the trouble codes with a ODBII. Nothing else present. The inverter itself seems to be functioning from what I can tell. Don't think it's the new battery. The car, as it stands, currently has no warning lights on because I have operated the vehicle several times WITHOUT turning on the A/C and seemingly sucking down the battery voltage and they cleared. The car starts and runs with no apparent issues at the moment. I feel, however, with about 110% certainty, that if I go out and run the A/C or decide to the drive to the mall with the A/C on that by the time I get back, we'll be in triangle mode again.

I checked the reservoir and there's coolant present. Could I see any turbulence in the reservoir with the car going? Nothing noticeable other than some vibration I attributed to he car being in operation. Had spouse and my friend check, too, to the same result. Indiscernible.

Does this, indeed, sound like the pump might be having an issue, and in turn, causing the inverter to overheat? Is there a connection here with the A/C other than it potentially pulling just enough more on the voltage?

I'm going to make a decision tomorrow regarding the next step but I wanted to get help from you guys. If you have questions, I'll try to respond as quickly as possible. Thanks!