How can I parametrize the address of docker registry in COPY --from argument?

I can't seem to be able to use a $VARIABLE reference in --from part of COPY instruction

$ docker build - <<<'
> ARG DOCKER_REGISTRY=docker.io
FROM $DOCKER_REGISTRY/library/docker:24-cli

ENV DOCKER_REGISTRY=$DOCKER_REGISTRY
COPY --from=$DOCKER_REGISTRY/kcov/kcov:latest-alpine /usr/local/bin/kcov* /usr/local/bin/
'
[+] Building 0.0s (2/2) FINISHED                                                                                                                                                                                       docker:default
 => [internal] load build definition from Dockerfile                                                                                                                                                                             0.0s
 => => transferring dockerfile: 243B                                                                                                                                                                                             0.0s
 => CANCELED [internal] load metadata for docker.io/library/docker:24-cli                                                                                                                                                        0.0s
Dockerfile:6
--------------------
   4 |     
   5 |     ENV DOCKER_REGISTRY=$DOCKER_REGISTRY
   6 | >>> COPY --from=$DOCKER_REGISTRY/kcov/kcov:latest-alpine /usr/local/bin/kcov* /usr/local/bin/
   7 |     
   8 |     
--------------------
ERROR: failed to solve: failed to parse stage name "$DOCKER_REGISTRY/kcov/kcov:latest-alpine": invalid reference format

Is it supported? There is nothing in the documentation suggesting you can't use build args there.

How else can I parametrize the registry there? (We have an internal proxy of docker hub, that I don't want to hardcode)

The ENV statement doesn't make a difference, I added it just in case it would.