Do I need to seal / moisture-proof an outdoor plug?
I don't think I'd worry about it.  I don't know much about UK electrical hardware but it looks like your outdoor plug / receptacles look a lot sturdier than the US equivalents.  (Apologies if I confuse any terms translating electrical terminology to English.)  

It looks like you have RCD / GFCI protection on outdoor receptacles / outlets, which is probably more important than the plug.  

The BS 1363/A plug is made so the cord exits down rather than straight out of the wall in most US cords.  This makes your outdoor cover a lot more low profile and probably a lot more secure than the US versions.  

The seam on the rewirable plug is probably less susceptible to water than the mating between the plug and the socket.  So you're really relying on the waterproof cover even with a molded plug.  

The only alternative I can think of would be to splice on a molded cord.  I am sure you can make a very secure waterproof splice.  But I personally would worry more about the splice than the rewirable plug.  

I will admit that sometimes I still add a wrap of Scotch 88 tape on things like this, even though it probably isn't needed, just because it can't hurt.  (The US electrical components may not be as robust as the UKs but we make the best tape in the world, hands down!)