The truss rod will be fine, but that tuning will be pretty useless, as a person who sounds like they could stand to learn a little more theory, I would keep the bass tuned to 4ths and learn the notes on the fretboard like normal, it is much easier that way.
Once you tune the D-E you have a minor 7th, which just changes something that is already nice and simple.
If you really want to do it go nuts, but if you were my student I would be strongly opposed.
What are you doing that your breaking strings? I have played extensively for a decade and have never once broken a bass string, mind you I don't do crazy pop and slap, but even then I couldn't see myself busting a string.