I'd suggest upgrading the base image in the Dockefiles to the current Debian stable (bookworm). There are both rust and debian slim docker images available for bookworm.
If you want to stay on bullseye then an alternative could be to use bullseye-backports. There is an official debian docker image based on bullseye-backports, however there is no bullseye-backports-slim: https://hub.docker.com/_/debian/tags?page=1&name=bullseye
However there are no bullseye-backports docker images for rust. Evindently I do not know how dependencies between Debian releases and Rust work, so the existence of a "backports" variant of Rust docker images might be a non-issue.
Alternative 3: adding the bullseye-backports repo during docker build
The third alternative would be to add the bullseye-backports repo to /etc/apt/sources and install the qgis server from there.
Recommendation
Going with alternative 1 is the least work long term - at some point you'll need to update to bookworm anyway so you just as well might do all the work now.
I'd estimate alternative 3 to be the least work short term, aka the quick hack.
I could certainly try to do a qgis_3.28_on_bullseye_backports branch if you'd think that would be helpful and the way to go.
I'm not sure I'd be able to sort out Rusty conflicts when - this is what I expect to happen - moving to a major new Rust version or a jump in the C-bindings (I'm guessing that this is the case) that come with bookworm.
Open questions
In case I'd go with the qgis_3.28_on_bullseye_backports branch then I have no clue how production ready the current HEAD is. Should I be forking on the v2023.03.14.01 tag then instead?
Could you please release a QGIS 3.28 version?
Alternative 1: bookworm images
I'd suggest upgrading the base image in the Dockefiles to the current Debian stable (bookworm). There are both rust and debian slim docker images available for bookworm.
Alternative 2: bullseye-backports images
Alternatively QGIS 3.28 is also in bullseye-stable: https://packages.debian.org/search?keywords=qgis
If you want to stay on bullseye then an alternative could be to use bullseye-backports. There is an official debian docker image based on bullseye-backports, however there is no bullseye-backports-slim: https://hub.docker.com/_/debian/tags?page=1&name=bullseye
However there are no bullseye-backports docker images for rust. Evindently I do not know how dependencies between Debian releases and Rust work, so the existence of a "backports" variant of Rust docker images might be a non-issue.
Alternative 3: adding the bullseye-backports repo during docker build
The third alternative would be to add the bullseye-backports repo to /etc/apt/sources and install the qgis server from there.
Recommendation
I could certainly try to do a qgis_3.28_on_bullseye_backports branch if you'd think that would be helpful and the way to go.
I'm not sure I'd be able to sort out Rusty conflicts when - this is what I expect to happen - moving to a major new Rust version or a jump in the C-bindings (I'm guessing that this is the case) that come with bookworm.
Open questions
In case I'd go with the qgis_3.28_on_bullseye_backports branch then I have no clue how production ready the current HEAD is. Should I be forking on the v2023.03.14.01 tag then instead?
Feedback wellcome. Thanks.