Well, let's be honest..
I read the topic and thought :
"
Oh another one of those... go tell him to stop even thinking about it..."
Here I am amazed by the beauty of it...
Well done sir!
Now the bad part...
bicycle wheels are not exactly the best choice for rides like that, even soapboxes (aka gravity racers) can produce enough inertia to crash on wheels like that...
and even worse: the larger the diameter the easier they will collapse on side loads.
(say a narrow curve or precisely too narrow for your speed

)
So please be very very careful, it'd be a real pitty to crash that thing;
it's too georgeous!
You can camber them slightly; so that the spokes have an easier job of deflecting the forces.
Or what I'd do.. upgrade the wheels.
If you insist on that wheelsize a perfect replacement would be spoked Moped wheels with 16" rims
They do often come with a drum brake which is a nice convenient add on for your kart

Since you're in the UK, check for older MBK, Herkules, Simson, Kreidlers, Puch etc...
front wheels preferred
like this one:
http://www.kalaydo.de/kleinanzeigen/technische-geraete/vorderrad-16-felge-rigida-nabe/a/39132502/
(yes it's expensive, I'm sorry

)
If you cannot find them, at least get bigger spokes (2.4mm if you can) maybe the rims need to be drilled out to allow the larger spokes to fit,
but the additional strength is really helping
(BMX ordinarily come with 1.8mm spokes, rarely 2mm)
If the hubs fail to allow the bigger spokes... get different hubs; you'd just move the weak spot down to the axle (and the longer lever would make it worse)
A wheel like the one above (here's just the image)
should help staying safe;
on the red bull derby you should STILL STAY AWAY from the jump they always include in the track
I wouldn't convert to a live axle btw.. not for a gravity kart.. makes no sense at all, just makes steering a lot more difficult.
Instead, find Wheelchair hubs and bearings;
they're meant to be single sided and thus come with a larger bearing.
well that is, if you stick with BMX rims
'sid
PS don't put an engine on it.. an electric motor no bigger than 500W maybe (twice the power the sinclair C5 had) but nothing bigger than that IYAM.