Emulsifier: is it safe to cook mayonnaise?
The mayonnaise itself is only a stable emulsion in certain temperature ranges. If you are talking about standard mayonnaise (made with egg yolk), it will split when heated, and won't help your new sauce at all. 

If you are planning to use one of those vinegar-oil emulsions sold in supermarkets under the name mayonnaise, I cannot predict how they will handle. They may or may not split. But they are not in principle supposed to be heated, so you can experience weird results. And the emulsifiers in them are calculated to be in the proper ratio for the mayonnaise as sold, so there will liekly be too little of them to sufficiently emulsify a sauce when added to it. 

Mustard won't help you at all. It is only a physical emulsifier, so only works with limited success. The cayenne flesh is already doing some physical emuslification, if that's not enough, mustard won't make it better. 

All in all, if you plan to heat your mixture (why?) then mayonnaise is unlikely to help you as an emulsifier.