After migrating my domain to Route 53 I finally transferred my website to a new Uberspace host which supports Let’s Encrypt. You should be automatically redirect to HTTPS when visiting www.schakko.de.
The whole procedure took 2 hours, including setting up the new Uberspace, importing the existing databases and changing the DNS records. Most of this was straight forward as the Uberspace team has a really good documentation for this.
BTW: Route 53 sets the TTL for each DNS record to 300 seconds by default. In most cases, 1 day should be sufficient. More DNS calls means more to pay.