Yes, have Ripley say "I had a bad dream I lost my hair" in a tongue in cheek way and be done with it. One line. No elaborate justification for the dream sequence. Go the Marvel / Deadpool route and make a joke of it, and move on.
Corporal Hicks: Not keen on this idea! Don't need that kind of humour.
Corporal, on principle I agree that Alien 3 and Alien: R should just be ignored, but if they really had to mention it, I would consider the off-handed joke above, which could be a reference to any other jokes Ripley may have had in cryo-sleep, to be the maximum extent of what is needed. I would prefer if they would just let it go and not mention anything.
I think I have a solution for everything. I have taken into account mostly what Cainsson said here, though keeping in mind what the skeptical Mr. Huda had said. To make the story make sense, there would have to be motivation for the cloning. I think I figured it out in a way that is neither convoluted, nor requires a retcon, and can in fact sync everything up nicely. So here goes, in point form.
-Firstly, we have to accept the premise that Bishop is operating under conflicting directives. Firstly, he is unable to harm a human being, but secondly, he has a specific request to secure an return an alien to company labs. But would Bishop consider a clone to be a human? Do you see where I am going with this?
-So how did the eggs get on the Sulaco? Simply, when Bishop was sent into the tunnels to get the second Dropship from the Sulaco, he went and secured 3 eggs from the egg chamber before the second dropship arrived. At this time he knew that there were only 3 survivors. He packed them neatly on the Dropship for return to company labs. At no time did he intend any malice towards any of the humans though, because since he was a scientific android sent on a scientific mission, he had a lab on board the Sulaco where he could store and secure such organisms.
-Once everyone is sleeping, Bishop wakes up and gets to work. But he is a bit of a spaz now that he is damaged. During the weeks of sleep that everyone is under, he clones Ripley, Newt and Hicks for the purpose of incubation. This does not conflict with his prime directive, because they are not humans. He could just deliver the eggs whole to the company labs, but he is concerned that they may hatch and screw things up for everyone, so he figures the best bet is to have the eggs get fertilized inside the clones, so that they can be frozen for further study once he arrives back at company labs.
-He successfully impregnates CloneRipley, and while he is delivering the second egg, being a spaz, he falls over, it hatches, and just goes for Clone Hicks on its own but it cracks the glass, hurts itself, and the blood starts the electrical fire that destroys the ship. This also explains why we have the burn marks inside the capsule from the acid. The clones are jettisoned along with Bishop to Fury 161, along with data on the cloning methods used. Perhaps Bishop was funneling DNA directly from the originals and that is how memories got included? I know this bit is a stretch. So we have a dead facehugger that causes burn marks in the capsule, an impregnated Ripley Clone, Clones of Newt and Hicks, Bishop plus 1 unopened egg in the capsule, + the data. When Bishop is reactivated in Alien 3, he sees that his Ripley clone is conscious and has all her memories and now that makes him guilty of violating his prime code. So he asks to be deactivated. The other egg hatches and impregnates the dog or ox. Hicks and Newt clones never once had an y consciousness.
-In Alien Resurrection, the marines are still trying to use Bishops original data for cloning purposes. In fact the previous point explains exactly why they went the cloning route, and how Ripley 8 was able to have memories. They never bothered to clone Hicks and Newt because they had no Alien DNA associated with them.
-And that leaves the 3 remaining survivors from Aliens who are also jettisoned in their own EEV, but without Bishop. That will deserve an explanation. Then they are floating out there until Alien 5. They are picked up eventually after they age about 30 years in stasis. Ultimately, the theme of Alien 5 could be about them learning what happened. Perhaps they could even end up frozen again after Alien 5 so that Ripley meets Ripley 8 eventually..