stick with firefox and download the addon "userAgentSwitcher" for its maturity and stability. Use the menus to emulate Safari
Revisit the site and verify it can access the content in the manner you like.
Then get the addon YoutubeDownload, open its menus and enable the "save SWF files for FlashGames" or similar option since there wil be only one like it.
revisit the site and have her finsih the game as she likes or until she turns old enough to discover boys. Open YT downloader and you'll find a list of files ready to download, select your location and do so accordingly.
Flash content, espeically interactive flash content like games is often segmented into multiple files to reduce the amount of time you spend caching content for play, most flash legacy flash content except for streaming YT videos but swf game files like you're leeching (leech is a generic term like "junk bond" and not a characterization) is often segmented at milestones like book chapters and explains why your prior attempts to access offline didn't succeed.