There are two different questions here, one objective and one subjective. I will try to give my view, for what it's worth. Bear with me.
First, you are asking what is the publication history of discrete mathematics? (Even if I suspect you know this much better than I do). Well, originally there was no such thing as DM. If I understand the history correctly, classical papers like this one by Hassler Whitney (on coefficients of chromatic polynomials) were viewed as contributions to "mainstream mathematics". What happened is that starting maybe late 60s there was a rapid growth in the number of papers in mathematics in general, with an even greater growth in discrete mathematics. While the overall growth is relatively easy to explain as a consequence of expansion of graduate programs, the latter is more complicated. Some would argue that CS and other applications spurned the growth, while others would argue that this area was neglected for generations and had many easy pickings, inherent in the nature of the field. Yet others would argue that the growth is a consequence of pioneer works by the "founding fathers", such as Paul Erdős, Don Knuth, G.-C. Rota, M.-P. Schützenberger, and W.T. Tutte, which transformed the field. Whatever the reason, the "mainstream mathematics" felt a bit under siege by numerous new papers, and quickly closed ranks. The result was a dozen new leading journals covering various subfields of combinatorics, graph theory, etc., and few dozen minor ones. Compare this with the number of journals dedicated solely to algebraic geometry to see the difference. Thus, psychologically, it is very easy to explain why journals like Inventiones even now have relatively few DM papers - if the DM papers move in, the "mainstream papers" often have nowhere else to go. Personally, I think this is all for the best, and totally fair.
Now, your second question is whether DM is a "mainstream mathematics", or what is it? This is much more difficult to answer since just about everyone has their own take. E.g. miwalin suggests above that number theory is a part of DM, a once prevalent view, but which is probably contrary to the modern consensus in the field. Still, with the growth of "arithmetic combinatorics", part of number theory is definitely a part of DM. While most people would posit that DM is "combinatorics, graph theory + CS and other applications", what exactly are these is more difficult to decide. The split of Journal of Combinatorial Theory into Series A and B happened over this kind of disagreement between Rota and Tutte (still legendary). I suggest combinatorics wikipedia page for a first approximation of the modern consensus, but when it comes to more concrete questions this becomes a contentious issue sometimes of "practical importance". As an editor of Discrete Mathematics, I am routinely forced to decide whether submissions are in scope or not. For example if someone submits a generalization of R-R identities - is that a DM or not? (if you think it is, are you sure you can say what exactly is "discrete" about them?) Or, e.g. is Cauchy theorem a part of DM, or metric geometry, or both? (or neither?) How about "IP = PSPACE" theorem? Is that DM, or logic, or perhaps lies completely outside of mathematics? Anyway, my (obvious) point is that there is no real boundary between the fields. There is a large spectrum of papers in DM which fall somewhere in between "mainstream mathematics" and applications. And that's another reason to have separate "specialized" journals to accommodate these papers, rather than encroach onto journals pre-existing these new subfields. Your department's "encouragement" to use only the "mainstream mathematical journals" for promotion purposes is narrow minded and very unfortunate.