Can an exotic species be also endemic?
I think this partially depends on the timeframe. Endemism is usually divided into paleoendemism and neoendemism (see e.g. Kier er al in PNAS for use of both terms). Here, neoendemism is referring to the result of recent adaptive radiations that have led to new endemic species. This is then in contrast to paleoendemism, which refers to previously widespread species that are now confined to isolated areas.

Neoendemism is clearly related to your question, but whether dingos should be labelled as neoendemic depends on if it can be seen as a unique subspecies. If it is, it might be more suitable to view it as neoendemic and not exotic (nothing prohibits neoendemic species to be ecologically "problematic" to the previous fauna though). 

In the literature (from what I've seen), neoendemism is usually used to describe historical events though (the result from adaptive radiations etc), so timeframe is obviously important (see e.g. Mishler er al, 2014). Even so, all taxa that has led to neoendemic species (through adaptations to local conditions) was by definition exotic at one point in time (e.g. when the most recent common ancestor of the Galapagos finches colonized the Galapagos islands). The same goes for the relatively recent adaptive radiation of Drosophila on Hawaii.

To me, neoendemic and invasive are not mutually exclusive terms, while the division between neoendemic and exotic may depend on time since colonization and whether the population now forms an independent taxon (e.g. subspecies) instead of a subpopulation still connected to its host population. I cannot find a reference or definition to support this interpretation right now though.