Some call it "chaos", others might call it "badly written software".
If you need to rebuild your application or even your Docker container (just bind-mount it in place and update the host, all applications fixed in one go!) then IMO you're bringing this kind of crap on yourself.
At most you should be reloading/restarting your application to apply the new timezone database, which shouldn't cause a problem if you've set up the right fallback and scaling systems to deliver whatever SLA you've promised to customers.
My shitty PHP frontends I built when I finished high school can update and reload the timezone database in less than a second through `systemctl restart php8.1-fpm`. If you need a month to update the timezone database, maybe you should've written whatever you made in shitty PHP just like me, because apparently shitty PHP is more foolproof in handling common occurrences like timezone changes than Enterprise Grade (TM) Cloud (R) Applications (C).