Q:

Time machine: include external FAT drive

I am trying to backup an external FAT drive with time machine on a new WD HFS+ formatted hard drive, and I've read it is possible. 
However, I find it in the excluded list, and I cannot remove it since it is grayed out. 
Do you have any hints? 

A:

Your FAT drive needs to be formatted using GPT[GUID] (or GPT+MBR which is how Boot Camp does it, but it's not necessary for a regular, data only, non-Windows OS disk) rather than 'plain' MBR. 
There are two utilities I know of which can change the underlying format from GUID to MBR without destroying the data - iPartition & the new Paragon Hard Disk Manager. 
Of the two, right now Paragon looks the most future-proof, iPartition hasn't seen an update in a while & cannot handle CoreStorage volumes.   
Not an advert, no affiliation, I'm not providing links you'll have to Google it
Right now, summer 2016, Paragon HDM is $40 but there's a company called BundleHunt - kind of a 'Groupon' for Mac apps - who are doing it in a bundle of any 7 apps for $16.  The resultant version you get registers properly with Paragon as though you'd bought it direct. 
Failing that, you would need to wipe & reformat the drive in Disk Utility - which cannot do the 'soft' conversion. 
It might be possible to do from terminal, but that's beyond my pay-grade, I'm afraid.