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After eating meat, there's a strong custom to wait a while before eating dairy, but does this apply if the dairy (or meat) food contained an amount of dairy/meat that, if it fell into the first food by accident, would have been "batul"?
For example, let's say that soon after I ate meat, I want to eat a food that is less than one-sixtieth dairy, and that dairy ingredient cannot be tasted (but was nonetheless mixed in deliberately). May I do so?
The seffer Maadanei HaShulchan in סימן פ׳ט s.v. 33 & 34 says that the תבשיל של בשר and תבשיל של גבינה which are subject to the minhag to be treated as real meat and milk as far as waiting goes, are specifically referring to תבשיל which do not have ששים and therefore do have טעם of whichever מין Is under discussion.
At a surface glance he does not seem to directly source this idea from earlier works.
But it seems that this is assumed. See Beis Yosef Orach Chaim 173 in the name of Rabeinu Tam who explains תבשיל של בשר as only having טעם. He also quotes הגהות מיימוני using this idea and saying therefore רוטב של בשר which is מושך טעם would be problematic. This point is brought in Taz in Yoreh Deah ad loc #5.
When explaining the concept of batel bshishim and ain mevatel l'chatchilah, the star-k gives the sources and explanations of these concepts. UNDERSTANDING BATEL B’SHISHIM (One-Sixtieth) However, the article deals with the actual mixing and whether or not it is kosher. The case given in the question could be derived from the analysis. Since the dairy (or meat) is not nullified, it is treated as if the second product is fully dairy (or meat). Thus, just as the mixture is asur, the second product must not be eaten within the forbidden time period.
However, based on the details of the case and the specific circumstances in the involved, there are various leniences that can be applied. However, the case as the OP describes it does not appear to have those leniences. For example, the description of "Dairy Equipment shows that while it can be eaten after a meat meal, it is not the case of the question.
The answers which deal with nonkosher also explain the circumstances that would apply with meat and dairy and when intensional bitul will apply. That is, just as putting a drop of milk into a meat pot would or would not allow bitul, eating a food with a certain amount of dairy in it after a meat meal would or would not be allowed.
However, the main discussion on this matter would seem to involve the DE marking on some foods. The star-k link could be followed if needed.
As seen in the dairy equipment explanation from the COR, it is only allowed if there is no dairy whatsoever in the food being considered.This implies that if there is anything dairy in the second food (which would not be batel l'chatchilah because it is a deliberate ingredient) it cannot be eaten after a meat meal and would not be batel.
As stated above, the pareve designation demarcates a product as having neither diary nor meat ingredients. The DE designation which stands for “Dairy Equipment” means that a hot product, while pareve in its essence, has been manufactured using equipment that was also used to manufacture hot dairy products where no kosherization occurred in between. It is important to note that a thorough cleaning of the equipment still occurs between production runs and there is absolutely no dairy products contained in an item labeled DE. Thus, from the standpoint of the kosher consumer, if someone has just completed a meat meal, he may then eat a product labeled DE. He cannot however eat something labeled DE during a meat meal. Some rabbis however are stricter in this regard and consulting with one’s own rabbi for personal guidance is always advisable.
As the star-k points out only items marked DE can be eaten in less than six hours and as the COR emphasizes, no item marked DE will have any dairy products in it.
Personally, from reading the COR description, I would consider the sources involving a dairy product or a nonkosher product to a meat dish (making the meat dish nonkosher if deliberately added in a batel amount) similar to the halacha of eating a kosher food that has a deliberately added dairy ingredient. Thus, the analysis of if the second dish can be eaten would be similar to whether or not the mixture is forbidden.
