Trent, if you really want to do it on the cheap and have a bit of time to spare, this is something you could easily do yourself. A mini is not exactly a car full of complex electrical wiring, it probably has one of the simplest wiring looms around.
You can buy rolls of wire in bulk, remove the old wiring loom, cut it open on your workbench and with the aid of a wiring diagram, replace each wire one by one, crimp on new connectors, solder each connection and then cover the bare ends with heatshrink. Bind it all together with PVC tape or stick it in a flexible loom tube and stick it all back in! Jaycar has all the stuff you need. From your photo I'm guessing your not going for 100% originality, so you could even do a new fuse box - aussie minis have two 35amp fuses which is almost pointless - most of the wiring in the car will melt with much less than 35 amps through it - which you now know!
In my opinion a secondhand loom is not really worth it, wiring that is 30 years old and has been subjected to engine bay heat over that time will be very hard and stiff, almost impossible to remove and reinstall without breaking it or having the insulation fall off - which will give the same result that you are already experiencing.
EDIT - The 'all black', debumpered look is *AWESOME*
