As you imply in your question, enzymes are catalysts, and as such, participate in a chemical reaction but remain unchanged after that reaction is completed. Therefore, once lactase has converted a lactose molecule into galactose and glucose, it is free to split another. According to this website, lactase can split up to sixty lactose molecules each second, so two lactase molecules can split 120; 4, 240; 16, 960. and so on. IOW, the more the lactase, the faster the conversion. This website provides a graphics that demonstrate how enzymes work.
So to answer your question, yes, there is enough lactase present in a lactose-free product to further catalyze any lactose present in any added dairy products. Whether or not there is enough to catalyze that lactose before it passes into the large intestines is another matter, however. When lactase is added to regular dairy products, they're allowed to "ferment" for several hours. Since lactose is not cheap, only enough lactase is added to catalyze a given quantity of lactose in a certain amount of time. My guess is that there will still be a significant amount of lactose present before it's passed into one's large intestines, where it causes the symptoms of lactose intolerance.
There is a lactase enzyme sold on Amazon that can be added to regular milk. It requires five drops for sixteen ounces of milk. If my calculations are correct, it works out to 0.8 ml per gallon of milk.
FWIW, the product is excellent. It makes a gallon of lactose-free milk about a dollar cheaper, which adds up.