The URL that triggered the error:
https://composr.app/data/endpoint.php/cms_homesite/personal_upgrader/_LEGACY_/?from=11.alpha2&supports_gzip=1&supports_zip=1&addon_actionlog=1&addon_aggregate_types=1&addon_apache_config_files=1&addon_authors=1&addon_awards=1&addon_backup=1&addon_banners=1&addon_breadcrumbs=1&addon_calendar=1&addon_captcha=1&addon_catalogues=1&addon_chat=1&addon_cns_avatars=1&addon_cns_cartoon_avatars=1&addon_cns_clubs=1&addon_cns_contact_member=1&addon_cns_cpfs=1&addon_cns_forum=1&addon_cns_member_avatars=1&addon_cns_member_photos=1&addon_cns_member_titles=1&addon_cns_multi_moderations=1&addon_cns_post_templates=1&addon_cns_signatures=1&addon_cns_thematic_avatars=1&addon_cns_warnings=1&addon_code_editor=1&addon_commandr=1&addon_content_privacy=1&addon_content_reviews=1&addon_core_abstract_components=1&addon_core_abstract_interfaces=1&addon_core_addon_management=1&addon_core_adminzone_dashboard=1&addon_core_all_icons=1&addon_core_cleanup_tools=1&addon_core_cns=1&addon_core_comcode_pages=1&addon_core_configuration=1&addon_core_database_drivers=1&addon_core_feedback_features=1&addon_core_fields=1&addon_core_form_interfaces=1&addon_core_forum_drivers=1&addon_core_graphic_text=1&addon_core_html_abstractions=1&addon_core_language_editing=1&addon_core_mediaelement=1&addon_core_menus=1&addon_core_notifications=1&addon_core_permission_management=1&addon_core=1&addon_core_primary_layout=1&addon_core_privacy=1&addon_core_rich_media=1&addon_core_themeing=1&addon_core_upgrader=1&addon_core_webstandards=1&addon_core_zone_editor=1&addon_counting_blocks=1&addon_custom_comcode=1&addon_debrand=1&addon_downloads=1&addon_ecommerce=1&addon_errorlog=1&addon_failover=1&addon_filebased_persistent_caching=1&addon_filedump=1&addon_forum_blocks=1&addon_galleries=1&addon_google_appengine=1&addon_health_check=1&addon_helper_scripts=1&addon_help_page=1&addon_import=1&addon_installer=1&addon_language_block=1&addon_ldap=1&addon_leader_board=1&addon_match_key_permissions=1&addon_newsletter=1&addon_news=1&addon_news_shared=1&addon_page_management=1&addon_phpinfo=1&addon_points=1&addon_polls=1&addon_printer_friendly_block=1&addon_quizzes=1&addon_random_quotes=1&addon_realtime_rain=1&addon_recommend=1&addon_redirects_editor=1&addon_robots_txt=1&addon_rootkit_detector=1&addon_search=1&addon_securitylogging=1&addon_setupwizard=1&addon_shopping=1&addon_sms=1&addon_stats_block=1&addon_stats=1&addon_syndication_blocks=1&addon_syndication=1&addon_themewizard=1&addon_tickets=1&addon_uninstaller=1&addon_users_online_block=1&addon_validation=1&addon_welcome_emails=1&addon_wiki=1&addon_wordfilter=1&addon_xml_fields=1

Lots of versions mentioned here

(Click to enlarge)

Your site isn't providing a "to" parameter to define what version you are upgrading to (I think this is normal)... and normally the site would just use the latest stable version if not passed. But composr.app doesn't have a "latest stable version" because v11 is still beta. Thus you get that error.

Try this.

Take the URL, put it in a web browser, and add "&to=11.beta2" after "from=11.alpha2". Run it. And it should download a TAR file. Use that in the upgrader for transferring files step (access the upgrader manually at baseurl/upgrader.php to avoid running into this error again).

I need to think about a solution. I don't just want composr.app to arbitrarily default to an unstable version.

NOTE: You may run into issues. I'd highly advise upgrading to alpha4 first, and then to beta2, so use "&to=11.alpha4" first.

My resolution was for composr.app to return an error instructing the webmaster to generate an upgrader manually from the site's news article about the release.