I realize I can use my Apple watch to ping my phone or use Apple's Find My app to ping my phone. But, I am looking for a creative way to ping my phone without the use of an app by simply using a no app based Bluetooth connection. Any ideas?
Just to make sure I'm clear, when you say "ping", do you want it to actually make a "ping" sound, show its location, or run a ping command to see if it has an IP address?
The answer to that won't really change my answer to this question, but I could be more specific.
The best way that I can think of to make your phone do anyone of these things without having to use either another device or computer, is to use RFID tags. You can buy programmable RFID tags that can be used to cause your phone to perform a task when in range of the RFID tag.
This will require some initial setup and, yes, an app to program the tag and to set the task for the phone to perform. There is no way, aside from witchcraft obviously, to make the phone do anything without an application to provide instructions for the phone to follow.
That being said you can set this all up and then just simply carry the RFID tag in an RF shielded container, so as not to set the phone off all day, and when you need to "ping" it, just pull out the RFID tag and get close enough for the phone to pick up the tags signal. Which if you do a cursory scanning of the first dozen or so results of you favorite search engine can range from literally touching, to "600 feet", that last one seems a bit exaggerated but I suppose it's possible. Or you can just use your watch.
When I can not find my phone I use a different phone to ring its number.
If I can not hear it ring I have to use my brain to remember where I last knew I had it and see if it gives me a good indication of where to start looking or repeat the exercise.
While old fashioned (it worked already with the first mobile phones) it still works.