According to Nintendo's Wii U to Wii U transfer FAQ:
Will I need to redownload content on the target Wii U after the transfer has been completed?
During the system transfer process, any content that was downloaded on the source Wii U will be added to Download Manager on the target Wii U. This means it will be queued to download automatically after the transfer has been completed. The maximum number of queued downloads is 64. Any excess downloads will need to be downloaded by adding them from the Nintendo eShop.
So, by using the Wii U to Wii U Transfer Tool, you should be able to redownload the pre-installed game. For safety's sake, I would recommend doing what you can to ensure both systems are linked to your Nintendo Network ID first, which may include calling Nintendo support; also, since the bundled copy of Splatoon is provided via download code, just download it after the transfer is done.