Update package.json versions as part of build step (#20579)

Fixes issue in the new build workflow where the experimental packages do
not include "experimental" in the version string. This was because the
previous approach relied on the RELEASE_CHANNEL environment variable,
which we are no longer setting in the outer CI job, since we use the
same job to build both channels. To solve, I moved the version
post-processing into the build script itself.

Only affects the new build workflow. Old workflow is unchanged.

Longer term, I would like to remove version numbers from the source
entirely, including the package.jsons. We should use a placeholder
instead; that's mostly how it already works, since the release script
swaps out the versions before we publish to stable.