The truss rod should not be used to adjust action. That is the job of the bridge and If you use the truss rod then your trading proper neck bow/back bow alignment for string height and may result in buzzing frets. The primary way to lower your string action is to use a neck shim.
Stew Mac sells them in precut shims in various sizes and they don't cost much. The second best way is modify or change your bridge.
By the way, a proper neck shim is not likely to have any impact on tone. Besides, if a shim does alter the tone, wouldn't luthiers and players be using them to adjust tone of their instrument? I was watching a bass setup clinic on YouTube by Roger Sadowsky and in summary, he stated that "a properly installed neck shim will have no effect at all on tone. Using a neck shim is the best way to adjust string height when you reached the lower limit of bridge adjustment.".
This is a Q&A on bass setup by two Master Luthiers - Rodger Sadowsky and Michael Tobias (MTD). Rodger briefly explains neck shimming at approximately the 13:15 timeline.
If you don't to use a shim then cut a deeper V in that one saddle as previously suggested. But if you do that, take the saddle off and put it in a vise to avoid accidentally scratching your brand new bass and to keep the metal shavings from sticking to the pickups and getting into places that can cause problems. If you file it while on the bass then at least completely cover the pickups with masking tape.
Bringing your bass back to original is simply a matter of removing the shim or replacing the saddle.
It's true the traditional bass is designed for standard tuning and standard string sizes. However, you'd think that after 50+ years of Fender basses that Fender would have added room for a couple more turns of a screw to accommodate alternative tuning and string size. Especially, when it would have absolutely no negative impact and could avoid having to use a shim in the first place.
The nut only effects the string height and sound of an open string. Once you press a string down with your finger that will then effect the string height and pitch.
This is why it's proper to use a capo or press down the first fret when setting string action.
However, replacing the nut is recommended since the nut has been cut for a G size string rather the smaller C. But before replacing the nut you can try filing it a little deeper with the correct size file. Your string will of course be lower but it just might work for you and may be able to avoid a nut job (pun intended).