Lightsail: Resolve private hostnames

Setup

There are multiple instances in our Lightsail account. They run Ubuntu 20.04 and their hostnames were set to backend (IP 172.26.0.10) and database (IP 172.26.0.11).

Problem

backend can reach database via ping 172.26.0.11 but it cannot reach database via ping database.

Question

Is there any way to configure AWS Lightsail to allow private hostname resolution?

We have tried:

• mDNS. Not possible because Lightsail prohibits broadcasts & multicasts.
• Route 53. We're not experts here but we tried to configure a private hosted zone. These zones need to be associated with a VPC but apparently we could only choose our default VPC which is different from the (invisible) Lightsail DNS.
• /etc/hosts. This would be our fallback solution if we cannot find an easier-to-maintain way to use private hostnames.