I had this same problem. Luckily I found it pretty easy to port FatFS over the the STM32U5. Just create a a new project in CubeMX using a microcontroller that *does* support FatFS. In my case I created a project using the STM32L4R9ZG. Then I copied all the FATFS files and directories over to my STM32U5 project. From there its pretty much a simple matter of changing all the header file references from STM32L4R9 device to STM32U5.
For my project I'm using a STM32U575RIT6 with a SD card interface. I was able to get everything working with FatFS and the SDMMC bus in 4 bit mode