If that's where your heart is, go for it. There are no wrong answers here.
It's worth noting a few things, though. Adding a 5th string adds a lot of tension onto the top, and the top ends up being thicker to account for it. The string spacing is altered, and the A ends up sort of in the middle of the bridge, which can we be weird to play. Obviously, fivers are common in Europe, so there are basses and bassists for whom it works great, but it's a bigger compromise, in my opinion, than simply adding a 5th or 6th string to an electric bass.
As a further counterpoint, since I'm playing devil's advocate, I have a bass with an extension. I love when I get to use the low notes in the orchestra, and I can sometimes use the open C to "cheat" notes in upper positions (there's a trick in the prelude of the 1st Bach suite where using your thumb to play a C harmonic at the octave). However, I often find myself longing to not have an extension at all. They're heavy, adding weight at the top of the instrument, and playing on the extension can be awkward and difficult. If I wasn't playing regularly with orchestras where the other players mostly don't have extensions, I would take mine off. I dream of having a "solo" bass where I can't unabashedly have no extension with no regrets.
One of my teachers (who was admittedly more of a soloist than an orchestral player) doesn't have an extension on his Laborie bass, and instead tunes to DADG when he needs a lower note. Tuning a normal lengthE to a C doesn't work so well, but I have done it. I know other people who have learned to just live in DADG for orchestra work.
Anyway, food for thought.