Man that is so disappointing. Sorry to hear that your little buddies can't come across with you. This should probably raised as a bug.
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There's a hard limit on how many eggs you can have in one save. They probably restricted transfer for eggs from expeditions to avoid going over this limit.
It would be worth raising anyway, as they may able to allow the transfer but not allow collecting them in the main save until you make room. Or allow transfer when the main save is below the limit.
But eggs != companion though. You can have an egg and just not allow hatching it if you dont have a free companion slot. How does it handle you redeeming a companion egg from expeditions or twitch rewards when all your slots are full. Presumably it functions exactly the same.
I feel this was more of an oversight than anything else
You would think so, but it won't let your companions lay more eggs when you're at the limit. This is separate to the companion slot limit. I assume it doesn't let you redeem eggs when you're at the limit either.
If I had to guess, eggs require more data than a typical inventory item, due to the complexities of creatures. They're probably implemented differently with some hacks to make it work, and limiting numbers is the simplest way to avoid complications.
That could well be it. Although I doubt it would require much data. We are talking about significantly less than terrain editing
The game has a lot of strange limits like that. Like allowing only 10 Message Modules, where Minecraft worlds allow infinite signs (or at least such a high number that, when you search how many are allowed, you can't really find an answer). Or only allowing you to store 16 Personal Wonders, even though they're just stored seed values, which are also about as simple as storing a text string.
Good to know
Likely the only way to make it work would be to use your main save to head back to the planet you got the eggs from in the first place