Jenkins doc build fails because of nested custom messages which are hosted in external, unreleased package

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I'm not entirely sure, but:

Would it be sufficient to fully release the asr_msgs package so it will be available as a binary package and could be found by rosdep?

yes, this would most likely solve the problem.

Or is there some other way to provide the asr_msgs package to the documentation build process?

There might be, at least according to this sentence in REP 141: ROS distribution files - distribution file (emphasis mine):

depends: list of repository names. Other repositories to perform cross referencing in the documentation. This is only necessary if the packages from the dependent repositories are not released. (default: [])

So a doc entry for a repository can have a depends key which lists the repositories the job would depend on. An example is shown in the same REP:

repositories:
  ...
  genmsg:
    doc:
      ...
    ...
  roscpp_core:
    doc:
      ...
      depends: [genmsg]

here roscpp_core's doc job would depend on genmsg.

There are a small number of doc jobs that use this, so it's likely this works (multiwii is an example: here, it depends onmsp. I'm not sure why it is configured that way, as msp appears to be released, but perhaps depends was used before it was released and never removed).