Hi Ian,
Welcome to the world of woodturning. I will answer your question about your bowl splitting with another question but please don't take it the wrong way if you already knew this.
Many people assume that because trees are round and bowls are round, that a bowl is turned from a slice through the trunk. Was that how you created your bowl blank?
Unfortunately the wood cut that way ends up with all the growth rings concentric and this is the worst case for splitting, as wood shrinks most circumferentially.
A bowl is much better turned from a section of the wood cut lengthwise from the log, so the grain is across the bowl. This is hard to describe but I hope that makes sense.
Regarding drive centres to fit in a chuck, I agree with Malcy that steb centres are very good but they ARE expensive, and they aren't a cone centre. If they slip the multiple teeth will tear the wood and you need to put more pressure on them to maintain drive. By the way, other manufacturers are producing very similar but less expensive, products.
You can get four prong drive centres that will fit in a chuck, (Nova do one for around £15 - often on sale on auction sites) but again this isn't a cone centre.
But I wonder if your reason for wanting a cone centre is wrong. Their key benefits are they will not try to split the timber, as may happen with the centre of a basic drive centre, and, second, they don't go far into the timber, good if you have expensive timber and are looking to reduce waste at the ends. I agree cone centres will slip if you are too heavy with the tool but that should not be the reason for wanting a cone centre. They don't make turning any safer, in fact I will put forward an argument that they are LESS safe as it is easier for the wood to come off the lathe, since there is very little metal in the end of the wood. And I totally agree with Lazurus's comments on this topic
Sorry for going on a bit, but hope that helps.