sleep eyes / rocking eyes / eyelids?

I think the main problems are the over all sculpting style and the upper lashes.

Most BJD have a sculpted eye lid. Simply adding a secondary lid inside the socket would likely look off on most sculpts. Dolls sculpted to have moving eyes usually have a more simplistic sculpting style and the eyelid is omitted entirely to make ways tor the moving eye lid.

On a BJD the upper lashes are usually glued to the sculpted eye lid, so if you added sleeping eyes, you'd either have to skip the lashes all together, or attach them to the moving eye or eye lid- baby doll style.
If you wanted to do removable eye lids you'd either have to skip the lashes, or make an open eyed version as well as any stage of closed you'd want and add lashes to each of them. This would also limit your face up options somewhat, as the complex shading sometimes used to give depth and life to the eye area may have to be simplified to account for the different lids (like, on an ordinary open eyed sculpt, you'd typically empathize the lid crease quite a lot, but when the lid is closed there shouldn't be much of a crease at all.)

That is probably why it isn't done a lot.

With that said, nothing is impossible.
You may need to do some modding to the eye lid or chose a stylized doll that has a very minimalist sculpting style.
If you want to use the classic weighted style of sleeping eyes, you'd need to figure out a way to make sure the S-hook isn't getting in the way, so perhaps replacing it with a low C-hook or a bar.
Also, if you want to go with the simplest style of weighted eyes, where both eyes move together, you need a sculpt that has both eye openings facing straight forward. Otherwise you need individually weighted eyes, but that could be done.
The space shouldn't be a problem, just scale the mechanism accordingly to the space.

 

I’ve modded eyes open and closed, but you kind of have to pick one or the other with BJDs because they’re static. If you were to do this, you would have to carve off Ida’s very prominent eyelids and then she’s going to look very O_O shocked by everything when her eyes were open. Otherwise you’d be adding a third eyelid to her underneath what she already has like a cat. It wouldn’t look like a human closing their eyes.

Blythes have a full eye mech you could check out to see if there’s a way to adapt something like it, but I feel like the extensive modding you would have to do to the doll would make her very much not herself by the time you were done. I guess it depends on how willing you are to drastically change her expression and whether you can live with it if you fail and she ends up half-modded.

 

I think most folks rather get a sleeping head/faceplate because it looks aesthetically nicer and seamless - (semi)realistic bjds have a lot of sculpting in the eye area which makes a babydoll or Blythe-like eye mechanism stand out a lot, especially if the eye holes need to be modded round. It'd probably be easier with anime-style dolls with less detailed and simply shaped eye holes.

I think the least invasive option would be to use clay to sculpt extra eyelids that can be attached with putty and easily taken off. Rocking eye mechanism usually looks as if the eyelid moves inside of the eyewell so it can look kind of disturbing inside more realistic faces Putting something on top would perhaps look more anatomically correct.

 

I have a sleeping head for Cooper. And the partially lidded eye version. And the open eye version.

Honestly? I think buying another head is a lot easier and maybe even cheaper than trying to mod a head to be able to open and close its eyes. More realistic, too. Even if there is no sleeping option and you have to commission somebody to mod one for you.

Look at this boy! He's so beautiful, why wouldn't I want three versions?? (Ok, he miiiiight actually have 8 heads, but I'm trying to stay on topic here. )

 

In 2010, Elfdoll introduced a sculpt "pinup girl Barbara," with separate, installable eyelids. There were two pairs of thin eyelids that attached with clay-- one pair for open eyes, and another pair for half-closed eyes. Apparently installing them without crushing the lashes could be a bit fiddly. If you search "Elfdoll Barbara" on the forum you will see some pictures that are still linked. Also some pics on Flickr. The sculpt has very simple eye sculpting to accommodate the eyelids. Also, be warned-- the smiling version of this sculpt is uncanny.

Edited to add

You could do similar to add lids, but as @Lillith has pointed out, the weighted mechanism for sleeping eyes takes up quite bit of space and has to be able to move freely. My baby dolls from years ago always had their eyes set back just a smidgen to allow rotation of the eyes and it bothered me (as a discerning 5 year old!) that it didn't look quite right.

 

A pet doll was released a couple months back with moving eyelids, can't find the additional photos now which showed inside the head, but will be around on Chinese sites still though like Weibo.

Christina on Instagram: "#newdoll #newrelease Time Limited #bjddoll , will close at 5th January. I put it at my Site-2: newcloversinging.com This is the small pet doll: Grey Sickle, made of resin, from #hatcat studio. It is mini #yosd size, about 4.5cm height. The sale price is $38 at my Site-2. The eye will be gift to you~ Free Shipping We offer Additional 4% OFF for old buyers welcome to mail me Payment methods: 1. PayPal and Bank transfer (5% OFF) are both OK. But we will ask charge for LAYAWAY. 2. After your order, I will mail you and send the invoice via PayPal. #bjd #bjddoll #bjdphotography #bjdphoto #bjdinstagram #bjdhobby #bjdsale #bjdstagram #bjdsales #bjdsalespost #bjdsofinstagram #balljointeddoll #legitbjd #minidoll #minibjd"

 

I love this idea of movable eyelids and have been trying to mentally puzzle it out and how it might work myself. I was thinking to have eyes made that are spheres and have the lid painted on the actual eyeball so that is rests close to the pupil and that way you could just reposition it in the head you might even be able to attach lashes to the eyeball at the edge of the painted lid.

 

And another head with moving eyes I had forgotten this release until I came across it again

zc on Instagram: "能稍微睁眼，不喜欢眼皮插件可以扣下来#bjdsculpting#bjdmaking#bjdhead#bid#bjddols# 寺仓自制#bjdphotofbjdmaker"

 

Oh, this one is pretty cool! The sculpted eyelashes create a really nice effect. Good balance between realistic and cartoony.