Thanks for your update.

For sake of research, I don't think being a laminate top ukulele makes any difference to how the ukulele sounds like with a pickup.
Pickups use mainly the vibration on the saddle for most of its tone, and a solid top would not make a big difference.
Fully "electric" ukuleles don't even have a top-wood, and are literally just chunks of wood - and they sound the best through an amp.

On an inexpensive uke like a KA-C, I would have gotten this pickup:

Pair one of these ARTEC undersaddles with an endpin jack on just about any ukulele, and you'll have a decent passive pickup option.

You can pair these with a preamp + batteries as well, but that makes the project a bit more complicated.

This Headway Snake brand pickup is an interesting find. I may keep it in mind for future projects