Apart from the obvious compatability advantages of a newer Kickstart are there any advantages with using the hard drive?
Well, essentially, the compatability advantages of the newer kickstart ARE the reason to upgrade to the newer kickstart. There isn't very much software that is HD installable that works under kickstart 1.3. Personally, I'd go for the 3.1 for it. 3.1 has almost identical backward compatability as 2.04 had, plus it supports all the newer things you want, like WHDLoad, etc. :-)
An AmigaDOS 1.3 system booted off a hard drive is a great novelty, and boy does it fly... But it's not very helpful to actually doing or playing anything. There's hardly anything that can be run that way, outside of Workbench 1.3, itself.