A battery is DC, definitely not the test instrument to use on a raw tweeter versus a midbass or woofer. If it wasn't damaged before it would be now. Also, a passive highpass filter on a tweeter will not pass DC from a battery.
I would get in touch with Odin ASAP and start by getting another tweeter diaphram on the way to you.....if you feel comfortable with doing this yourself (Odin can walk you through it). If not, then get the speaker back to Odin and let him handle the diagnosis and warranty.
Btw, getting the tweeter diaphram back for inspection is important in determining the cause.