If a fully charged (allegedly) battery won't start the bike at all, then it's pretty certain that it's knackered, and it's almost certainly going to be the 17-18V from the reg/rec that did it. Before spending £59+ VAT at DSS for one, it's worth taking a look on eBay. DSS don't seem to carry the Honda OEM one, but are virtually charging Honda prices for an aftermarket part.
Don't buy a secondhand genuine one as they usually lack fins which helps them on the way to destruction as they overheat, but consider some of the offerings on eBay. Dirt cheap is almost certainly going to be trouble, but middle of the road should be fine. I got one of these from this seller
http://www.ebay.co.uk/itm/HONDA-HORNET- ... 460787edf6 the best part of two years ago, and it's behaved impeccably. You can also fit one from certain models of Yamaha R6 with a little work. I think you have to break the connections out of the connector and fit each pin individually, but if you do a search on the forum you'll find several threads on fitting one, plus other make/models. There are lots of threads on here with people's experiences of using aftermarket reg/rects like the one above, R6 ones and others, so I'd have a little search about and decide what you want to do before firing almost £80 at DSS for one if you include postage.
As regards the thermal stuff. Due to the way the things work, the reg/rec gets very hot in normal use as heat is a by-product of the manner in which it converts AC from the alternator into DC for the charging circuit and regulates the output to the correct level. Excess output is effectively turned into heat. This means that keeping it cool is vital to keeping it happy. Excessive heat can cause electrical components to run out of spec, and eventually lead to them breaking down and not functioning correctly even when at the right temperature. Think about why you have aircon units in server rooms, a big heatsink and fan on the processor in your PC, etc. Hence why even on a cool day, a knackered reg/rec will still not work properly, in your case giving too many volts to the charging circuit and damaging the battery. In extreme cases, they have been known to catch fire!
So, you need to help keep it cool. The unit is mounted against a flat plate on the subframe, which effectively acts as a heatsink and helps conduct the heat away. In computers, and on other things with microchips and other components that get hot and need a heatsink, it is common to spread a conductive paste on them to ensure the maximum thermal conductivity between the component and it's heatsink. On my NC35, the plate on the subframe appears to have a very slight shape it it, so in theory there's a bit of a gap between the reg/rec and the plate in places, which means that it isn't transferring quite as much heat as it could. A coating of thermal paste could be used to help reduce or eliminate the gap or gaps, and increase the heatsink effect of the subframe. These days, you can also get sticky pads that do the same sort of job. Personally, I don't use the stuff, however the previous owner acquired a chunky aluminium plate to go between the reg/rec and the subframe, so the extra material will help. I've also seem people fit computer fans to the reg/rec.
You can get thermal pastes, adhesives and pads from somewhere like Maplin. Depending on the grade, it will cost up to a tenner for a syringe of the stuff, and you get very little for your money. However, there will be enough for what you want to do. The pads come in varying sizes and thicknesses. I've not used pads, so I don't know how much give they have, so if they aren't very squishy, they won't be much good for smoothing out any contours in the plate on the subframe.
http://www.maplin.co.uk/thermal-compound-351526
http://www.maplin.co.uk/thermal-adhesive-tape-346530
The above are simply suggestions, there are dozens and dozens of sources and choices for thermal pads, adhesives and compounds, so have a search on eBay, Maplin etc....