Your tow vehicle should power the brake lights, turn signals and running lights independently of the trailers battery. In fact the trailer battery should not be connected at all to those lights.
Just be aware that with the battery removed, you no longer have breakaway brakes...the battery is required to be aboard for travel, by law in most states, in order to activate the breakaway brakes should your TT come off the hitch... Just an FYI
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My 2 cents, your mileage may vary...