It is not clear, but I'm skeptical that the Fourteenth Amendment, in particular, would have been allowed under the Corwin Amendment, even if modified to exclude slavery.
The Wikipedia article you link includes this line:
The contentious debate in the House [right before they approved the amendment and sent it to the Senate] was relieved by abolitionist Republican Owen Lovejoy of Illinois, who questioned the amendment's reach: "Does that include polygamy, the other twin relic of barbarism?" Missouri Democrat John S. Phelps answered: "Does the gentleman desire to know whether he shall be prohibited from committing that crime?"
So, even in the moment, they were aware of this ambiguity, and chose not to clarify the text (also, their politics were just as polarized and toxic as ours are today). At a minimum, I imagine the Corwin Amendment would have been interpreted to include state laws in the immediate penumbra of slavery. We can conjecture that, if slavery had nevertheless been abolished state-by-state (which may or may not be wishful thinking), the Amendment might have extended to miscegenation laws, and by extension to the Fourteenth Amendment, which was used as a rationale for striking down said laws in Loving v. Virginia.
The Fourteenth Amendment was an unprecedented expansion of Constitutional obligations upon the states. Before this point, states had been widely understood to be semi-sovereign, in an arrangement not entirely dissimilar to the European Union today (but with many important differences). Given the context in which the Corwin Amendment would have been ratified, I find it hard to believe that a modified Fourteenth Amendment would have been Constitutional in this alternate history. But then, there might not have been the political will for it in the first place, if we imagine that the Civil War was prevented.
On the other hand, we should recognize the rather extraordinary level of circumlocution the Founders used around slavery. The word "slavery" first appears in the Constitution in the Thirteenth Amendment, when it is banned, yet numerous clauses in the original text refer to the practice indirectly. The Corwin Amendment was obviously mimicking this style. In fact, the Corwin Amendment as adopted was substantially more direct than the original proposal:
No amendment of this Constitution, having for its object any interference within the States with the relations between their citizens and those described in second section of the first article of the Constitution as "all other persons", shall originate with any State that does not recognize that relation within its own limits, or shall be valid without the assent of every one of the States composing the Union.
It is also possible that "domestic institutions" was intended more as a political rationale than as a concrete set of policies (i.e. it's just the "states' rights" argument in different clothing). If so, then it did not mean anything and was merely a fig leaf for the protection of slavery. Of course, after enactment, it might have been interpreted differently from the original intent.