Problem solved by creating an image of the old card with Win32 Disk Imager in Windows, as recommended here:
The image is approximately 8 GB in size – do I need to take anything else into account on the new card, given the significantly larger storage capacity, or is it sufficient to simply save the image?
The tool didn’t allow me to create a smaller image with just the actual data.
The size itself isn’t really a problem for storage space, the new card has 512 GB… 