Need to jump start every time I it shut down

Greetings Prius Gurus,

Have 2005 which I maintain myself basically (except for change of main battery which was done ~75k miles ago). I changed the inverter coolant pump, plugs, coils, and water pump and secondary battery (this very recently). Runs great for 240k. Problem is for some reason each time i want to drive it i must jump it at the fuse box. For some reason, for the last 4 months (don't use it that much - 2-3x per wk) nothing comes on when using the fob. Doors won't open, no dash lights, etc. Press start with brake - nada - even with fob in slot.. Have to jump fuse box every time with small costco portable auto battery jumper and must have key in slot each time after I turn it off.. My reckoning:
1) on-line replacement key fob I purchased is faulty? Installed new fob battery but no help. I think once right after I drove it and shut it down, I immediately used the fob it worked, no need to even be in the slot. but that was only 1-2x that it actually worked normally.
2) the "key" button under the steering wheel is wrong position? Think it has something to do with the alarm or auto-fob? Tried it in both positions but didn't provide remedy.
3) Something is staying powered on after shutdown and consuming battery energy?

Question: How can I see if in fact it is #3 above (most likely) something is sapping the battery despite turning car off? i have a lower-end Zurich hand-held diagnostic machine I use for reading, deleting errors codes. I think it has some electrical diagnostic capability as well. Think I can use that in this particular instance? Any pointers on what sequence of troubleshooting steps I should try? Or did anyone have any specific experience with this fault and give me a cut-to-the chase instruction?

As always, many thanks in advance. Love my old 2005! I also have a 2010, 2015, and 2017 Prime. Love em' all but as they say - nothing like the first time! 2005 got me hooked.