Take any lined kippah, snip a small hole in the lining. Coat the edge with fray check or fabric glue, let it dry, being sure the inner edge isn’t touching the underneath fabric, and then slip one side of the snap clip barrette into the hole.
What a cute idea. :) First off, it depends on how crafty you are and your budget. There's no shame in commissioning an artist for this, especially if you have a deadline to reach! I have not looked into such services, but I'm sure you could find someone on Etsy who would do this sort of work.
Of course, a DIY, even one that isn't super perfect, would be treasured (I certainly would prefer that myself were I the recipient). You can find doll heads and parts easily online (check out dollhouse supply websites) and those ought to carry outfits and clothing as well. As you know the scale, you can also check out places like Amazon.
If you want to do something even easier and that still has a really great "realistic" effect, I would suggest simply buying printable iron ons, fabric glue, and making more simple "plush toys." Using a stuffing of your choice and some pipe cleaners for an internal skeleton would keep them nicely posable.
As for an arcade cabinet, if it's in your budget, you can always buy tiny real working ones or make one by scaling down a printable (video guide included). I'm not entirely sure the best approach for a gondola, but I'm sure there's enough reference material online to DIY it should you go that route.
I hope at least some of this was helpful. Good luck!