Just some thoughts.
Breeding them together would result in quite the mish-mash. The two breeds were developed separately. In fact, the Whites were developed in Quebec and called Chanteclers, but the Partridge were developed by another fellow in Alberta and were originally called Albertans (changed to Partridge Chanteclers when accepted into APA). Of course, it's more complicated than that, but you get the gist.
The original Chanteclers were made up of breeds like White Leghorn, Rhode Island Red, Dark Cornish, Columbian and White Wyandottes, and White Plymouth Rocks. So the possibilities of what can be hiding under the white coloration of the Chanteclers is quite varied. You've got the possibilities of blue/black/splash, barring, mottling (from Leghorns), lacing (Cornish), red (from Reds), columbian pattern (Columbian Wyandottes). And the e Locus could be eb, eWh, ER, or ? (E Locus is the coloration base, so to speak, of all chickens.) It's a real grab bag. The white gene c/c, which covers all those colors and patterns with white coloration is recessive, which means that both the sire and the dam must be carrying c (white) for the chicks to be colored white. So crossing a Chantecler with a Partridge Chantecler will most likely give you brown chicks or possibly chipmunk chicks which grow up to be partially laced or mossy. Of course, it's much more elaborate and detailed than this, but it gives you some idea what you are probably going to be dealing with.
However, they would still have the great features that all Chanteclers possess. And with subsequent crosses, you will have sires and dams that both carry recessive white (c) which will produce c/c (white) chicks. So within two generations, you would have the white birds back again. I think some people wouldn't want to buy any of your White Chantecler chicks, though, since they would be sort of polluted genetically by the infusion of Partridge blood/genetics. But most people would never know (I am not advocating this, btw). And you could still show them because that's how showing in the APA works--if it looks like a Chantecler, then it can win ribbons as a Chantecler. Your Partridges, though, would probably look fairly messy and would probably not cut it in the show ring--especially if any of the Chantecler were carrying dominant white (I) which will show up as white even if only one parent is white at the mating.
I say, run them all together, but at breeding season separate them into to coops and breed them separately. It would only be for about 6 weeks. 4 weeks to clear the ovaries of the hens, and 10 days of collecting eggs for the incubator. Then they could all go back together. Fairly simple. If you wanted, you could mix breed a few after that and then you would be able to make comparisons, etc. The chicks would be younger than your first hatch and easy to keep separate for a few months, then you could mark them (if you hadn't banded them earlier). Publish results and observations.
Again, those are just some thoughts thrown out there. Some folks would definitely say, "Never mix them! It would be a travesty!" They might be correct.
