This is the solution I came up with:

import argparse

# See https://stackoverflow.com/a/8632404/2451238 + argparse._HelpAction source
def short_help(help_message):
  class shortHelpAction(argparse.Action):
    def __init__(self,
                 option_strings,
                 dest = argparse.SUPPRESS,
                 default = argparse.SUPPRESS,
                 help = None,
                 deprecated = False):
      super(shortHelpAction, self).__init__(
            option_strings = option_strings,
            dest = dest,
            default = default,
            nargs = 0,
            help = help,
            deprecated = deprecated)
    def __call__(self, parser, namespace, args, option_string = None):
        print(help_message)
        parser.exit()
  return shortHelpAction

ap = argparse.ArgumentParser(add_help = False, conflict_handler = "resolve")
ap.add_argument("-h", "--help", action = short_help(None),
                help = "show help message (common parameters only) and exit")
ap.add_argument("-H", "--help-all", action = "help",
                help = """show extended help message (incl. advanced
                       parameters) and exit""")
ap.add_argument("-v", "--version", action = "version", version = "1.0")
common_args = ap.add_argument_group("common parameters",
                                    """These parameters are typically
                                    enough to run the tool. `%(prog)s
                                    -h|--help` should list these
                                    parameters.""")
common_args.add_argument("-f", "--foo", metavar = "<foo>",
                         help = "the very common Foo parameter")
common_args.add_argument("--flag", action = "store_true",
                         help = "a flag enabling a totally normal option")
ap.add_argument("-h", "--help", action = short_help(ap.format_help()))
advanced_args = ap.add_argument_group("advanced parameters",
                                      """These parameters are for advanced
                                      users with special needs only. To make
                                      the help more accessible, `%(prog)s
                                      -h|--help` should not include these
                                      parameters, while `%(prog)s
                                      -H|--help-all` should include them (in
                                      addition to those included by `%(prog)s
                                      -h|--help`.""")
advanced_args.add_argument("-b", "--bar", metavar = "<bar>",
                           help = "the rarely needed Bar parameter")
advanced_args.add_argument("-B", "--baz", metavar = "<bar>",
                           help = "the even more obscure Baz parameter")
advanced_args.add_argument("--FLAG", action = "store_true",
                           help = "a flag for highly advanced users only")
ap.parse_args()

python extended_help.py -h

prints

usage: extended_help.py [-h] [-H] [-v] [-f <foo>] [--flag]

options:
  -h, --help       show help message (common parameters only) and exit
  -H, --help-all   show extended help message (incl. advanced parameters) and exit
  -v, --version    show program's version number and exit

common parameters:
  These parameters are typically enough to run the tool. `extended_help.py -h|--help` should list these parameters.

  -f, --foo <foo>  the very common Foo parameter
  --flag           a flag enabling a totally normal option

,

python extended_help.py -v

1.0

, and

python extended_help.py -H

usage: extended_help.py [-H] [-v] [-f <foo>] [--flag] [-h] [-b <bar>] [-B <bar>] [--FLAG]

options:
  -H, --help-all   show extended help message (incl. advanced parameters) and exit
  -v, --version    show program's version number and exit
  -h, --help

common parameters:
  These parameters are typically enough to run the tool. `extended_help.py -h|--help` should list these parameters.

  -f, --foo <foo>  the very common Foo parameter
  --flag           a flag enabling a totally normal option

advanced parameters:
  These parameters are for advanced users with special needs only. To make the help more accessible, `extended_help.py -h|--help` should not include these parameters, while
  `extended_help.py -H|--help-all` should include them (in addition to those included by `extended_help.py -h|--help`.

  -b, --bar <bar>  the rarely needed Bar parameter
  -B, --baz <bar>  the even more obscure Baz parameter
  --FLAG           a flag for highly advanced users only

.