I previously created a post describing a configuration that I believed was working, but it turned out not to be correct. Since then, I have made additional changes to my setup, and everything now appears to be working as expected. I would therefore like to share my current configuration, both in case it is useful to others and as a reference for myself should I need to set this up again in the future.
To begin with, I am not entirely sure how well SSH was working before, since most, if not all, of my repositories were configured to use HTTPS with Personal Access Tokens.
While investigating the issue, I realized that for Bitwarden to act as the default SSH agent, it should be configured in the same way as other third party SSH agents. After some searching, I came across a support page from Strongbox that explains how to configure a custom SSH agent using the IdentityAgent option in the SSH configuration file. Following that guidance, I created the file ~/.ssh/config with the following content:
Host *
IdentityAgent ~/.bitwarden-ssh-agent.sock
After applying this configuration, SSH started working correctly with the Bitwarden SSH agent. I hope this information is helpful to others who are trying to achieve the same setup.