Hi Rich,
yes all karts racing in MSA, now Msuk, run with both wheels keyed and bolted tight to a one piece axle (no diff). As you know when turning any corner the inside wheel will follow a different curve than the outside wheel, because they are fixed together by the axle, the kart will try to go straight, this causes the kart to push on (or under steer) There are a few things you can do to combat this, two main things are chassis design and adjusting the front end camber/caster, in my opinion caster is more important.
There is also a relationship between front end track width and back end width, for instance if the back has too much grip compared to the front that will also cause under steer. In kart racing we call this ‘balance’.
This is a vast subject which should include every thing from how the engine delivers power to driving style and where you place the kart on the track, especially if you only have 6.5 hp. Out of interest two stroke karts like Rotax produce about 29hp. F100 100cc about 34hp both weighing in at around 160kg
There are plenty kart axles out there that I reckon would work for a CycleKart, they range from 25/30/40/50 mm. 40 and 50 are hollow and come in a range soft to hard, a 50mm will bend easier than a 40mm, 30 mm comes in both hollow and solid, 25 mm are solid, all have key ways cut in them. Look at kart shops like Dartford karting/KKC karting.
Word of warning if you see one on e’bay it is probably bent, but you do see axles with other bits on it like brake disc and carrier and sprocket carrier, which may be worth buying.