Same wine, different bottle: starting issue

I posted about this problem a month ago but it's a bit different now, so thought maybe better to start a new thread.

My 2016 Prius V randomly will not start. It might be 19 out of 20 times it starts or five out of six. It starts immediately with my battery booster thing. Sometimes, if I pump gas pedal (per suggestion), it starts after maybe 2 minutes. Sometimes when I'm looking for the booster thing, it suddenly starts. No real pattern.

My mechanic checked the 4 yo battery and said it's fine. I was concerned about mice having chewed wires (caught two in car but no sign in air filter), but he said the lines to battery were fine (not sure if he really checked). Said he checked fuel and other sensors (suggested), all fine.

He wondered about the ev/eco/power buttons in console, which have always gotten stuck (bought used). When I diconnected it, the eco light stayed on!!! reconnected and go it off...but still have same problem.

My mechanic (who owns two Priuses himself) is stumped.

He wondered about key fob. Not sure what that would do to starting, but that
has been weird. For awhile, as soon as I approached the locked car with fob in purse, it would unlock on its own. If fob is far enough away, far remains locked. Replaced battery, used other fob, seemed to solve. This week l, noticed it happens when go to hatch, but not other doors. But again, don't see what fob would have to do with the car starting.

Anyone got any clue? Anything else we can try? Much obliged

 

I connect rhe batter booster thingy where the manual aays, under hood, in what looks like a circuit box, under a little blue cap.

 

(Mods--couldnt seem to edit my original post to put this up, appreciate if you can add)

UPDATE: went to Toyota place and talked to a mechanic. He said it was either the auxiliary battery, the hybrid battery or an electrical problem. My regular mechanic said the auxiliary battery was perfect, and the hybrid battery is only 7 years old so seems unlikely.

I have become suspicious of the power button. I have been trying to push it down more firmly and hold it down and maybe that helps? But I also noticed that often when I'm getting ready to turn it off after it won't start that when I push that button, the ready light suddenly comes on and everything's good. The more I try this the more consistent it seems to be.

Now I know just about anybody else would give up & just pay the $140 to Toyota just for the diagnosis, then who knows how much for their fix, but I just paid $350 for a new windshield, plus like trying to figure it out--with this group's help.