Thanks, but I will need full image to create a new working SD card, your backup only contains the first partition. From what I have found out there are 3 partitions on the card:
Disk /dev/sdd: 7.4 GiB, 7948206080 bytes, 15523840 sectors
Units: sectors of 1 * 512 = 512 bytes
Sector size (logical/physical): 512 bytes / 512 bytes
I/O size (minimum/optimal): 512 bytes / 512 bytes
Disklabel type: dos
Disk identifier: 0x00000000
Device Boot Start End Sectors Size Id Type
/dev/sdd1 * 8192 131071 122880 60M 6 FAT16
/dev/sdd2 131072 5373951 5242880 2.5G 83 Linux
/dev/sdd3 5373952 15523839 10149888 4.9G 83 Linux
the third partition looks like user data, I can see last .stl files I printed there. The one I need to recover is the second one that contains the system.
When I copied your backup to the first partition and used second partition from my damaged card, then connected monitor to the internal raspberry I can see that it starts booting, but then there are some segmentation faults and the printer fails to start properly.