Unable to increase Macintosh HD partition size with data in a subsequent partition I can't make my main Macintosh HD partition bigger. It's limited by the FAT partition in the middle, which seems to be frozen, I can't move it down to make space for my main partition.

To get some background, I originally had those 2 partitions sharing the space in peace. Then, I created a 3rd partition to experiment with another OS. Once done I deleted that new partition, and boom. It became a no-man's land that neither partition can claim.

I just want to have things back the way there were :(
It appears you have space to image the entire DISK0S4 partition to a compressed img file that would be stored on Macintosh HD. 

From there you will be free to delete the partition, upsize Macintosh HD on the fly, and then make a new DISK0S4 FAT.

To start, just select the DISK0S4 and then go to the menu:


New ▶ Disk Image from "DISK0S4"...


You'll want to mount and verify the image well before erasing anything. Also consider it will be at least twice as slow to move data from one part of the drive to another if you have a HDD with physical heads that have to move. Dumping that partition to an external drive is far faster - you can still use Disk Utility to compress the image to save space and time since the CPU can usually out compress any external disk at less that 25% usage - the bottleneck is the write speed of the drive.