Putting the ExtensionManagementUtility::addStaticFile() call into Configuration/TCA/Overrides/sys_template.php is correct,

but keep in mind to clear your system caches (the red lightning) afterwards.

Also keep in mind that $_EXTKEY is unavailable in that file, use your extension key in a string instead.

Example: \TYPO3\CMS\Core\Utility\ExtensionManagementUtility::addStaticFile('distribution_ext', 'Configuration/TypoScript', 'Distribution Extension');