I am looking for a console command (that can be run from Battlemetrics and other RCON) that can alter the config option "Backpacks drop on death" between true or false

Bonus: Make a global chat statement about the switchover.  Have this in the config such as:
announceDropChange: True

And it would print to chat something like "Backpacks now drop on death!" or "Backpacks no longer drop on death"

(Reasoning:  We run PVE servers with a PVP day.  This is becoming quite the popular setup lately too it seems.  But when we go PVP mode I have to manually edit this config so backpacks start dropping during PVP)

I will donate for this change.

You can just set the config option to true permanently (always drop on death), then use console commands to grant or revoke the keepondeath permission to the default Oxide group. You can also run a say command from the server to announce this. No plugin needed.

Not a bad idea.  But would still love a command built in to backpacks to do it.  Just makes things a lot easier than messing with permissions etc.

"Messing with permissions etc" is just a console command.

o.grant group default backpacks.keepondeath
say "Backpacks no longer drop on death"

o.revoke group default backpacks.keepondeath
say "Backpacks now drop on death"

More importantly, what you should be asking for is a way to automate this so that 0 manual effort is required to switch between PvE and PvP modes. You can incorporate the Dynamic Config plugin which can alter the config of Backpacks and reload it for you.