An ethical dilemma is a complex situation that often involves an apparent mental conflict between moral imperatives, in which to obey one would result in transgressing another.
Before considering methods to resolve the ethical dilemma you must first decide that there actually is one. It is by no means clear from your question what you think constitutes the "moral quandary": you have identified possible actions the party can take (including one that you have flagged that they "would not be COOL with") but have not told us what moral imperatives you feel are in conflict. If there is only one option that is morally unacceptable to you then you have no problem - simply do anything but that. However, reading between the lines, it seems that you have a problem with all of the options you have suggested but have not said why - so we have to guess.
This is my guess - tell me if I'm wrong:
Clearly, you are playing D&D so you have no moral issues with killing sentient creatures like the kobolds and cultists who are guarding these eggs. So, either you have a problem with killing black dragons in particular (which seems unlikely) or you have a problem with the fact that they are innocent sentient creatures - lets go with that as moral imperative No 1
If the party does not interfere in some way then the black dragons will hatch and in the fullness of time they will most likely follow their natures and become nigh on unstoppable killing machines - not allowing this to happen is moral imperative No 2
The Monster Manual tells us:
The black, blue, green, red, and white dragons represent the evil side of dragonkind. Aggressive, gluttonous, and vain, chromatic dragons are dark sages and powerful tyrants feared by all creatures- including each other.
They believe that the world's wealth belongs to them by right, and a chromatic dragon seizes that wealth without regard for the humanoids and other creatures that have "stolen" it.
Chromatic dragons are united by their sense of superiority, believing themselves the most powerful and worthy of all mortal creatures. When they interact with other creatures, it is only to further their own interests. They believe in their innate right to rule, and this belief is the cornerstone of every chromatic dragon's personality and worldview. Trying to humble a chromatic dragon is like trying to convince the wind to stop blowing. To these creatures, humanoids are animals, fit to serve as prey or beasts of burden, and wholly unworthy of respect.
The most evil-tempered and vile of the chromatic dragons, black dragons collect the wreckage and treasures of fallen peoples. These dragons loathe seeing the weak prosper and revel in the collapse of humanoid kingdoms.
All chromatic dragons are evil, but black dragons stand apart for their sadistic nature. A black dragon lives to watch its prey beg for mercy, and will often offer the illusion of respite or escape before finishing off its enemies.
So, in summary: not nice people.
However, the Monster Manual also says:
The alignment specified in a monster’s stat block is the default. Feel free to depart from it and change a monster’s alignment to suit the needs of your campaign. If you want a good‑aligned green dragon or an evil storm giant, there’s nothing stopping you.
But this is simply a specific instance of rule zero - if you as DM want's to do this, you can. So, do you want to do this? If "the needs of your campaign" don't require this particular ethical dilemma, don't do this: in your world these creatures are inimical to all PC humanoid species and killing them is a positive duty. "Nuke the entire site from orbit--it’s the only way to be sure" because you are actually dealing with creatures that are worse than this (and if you can make your players feel like Ripley feels in this scene kudos to you):
However, if "the needs of your campaign" require you to make your players squirm on the horns of an ethical dilemma then make sure they know that these creatures are currently not a threat to the party and that, with appropriate intervention, they may never become a threat to anyone.
Some options that I can think of are:
1. Kill them
2. Leave them a) without any kobolds they will probably fight until one survives which may or may not survive depending on natural predation b) with the kobolds there they would probably all survive c) and tell someone else about them (like Governor Nighthill)
3. Take them a) sell them to the highest bidder b) give them to a loving and nurturing family (of metallic dragons for preference) c) raise them themselves.
Some of these options are morally questionable and some are practically difficult - that's sort of the point. RPGs are about making choices, sometimes hard choices, and living with the consequences. It is worth noting that if you play this through to Rise of Tiamat, there are external consequences that flow for the decision the players make here.
The party I am DMing did this room just last week - you can see what they decided here (Spoliers! Duh!). The discussion grew quite heated and a good time was had by all, however, I realise that this type of intra-party conflict may not be a good time for everybody.