I have never worked with transmembrane proteins; therefore, my advice may be totally off the mark.

However, I worked with lipases, which are active at the lipid water interface. I just had to mix the lipase fraction with micelles and they would attach themselves.

With your transmembrane proteins it should be similar. They have large hydrophobic areas that should be attracted to the micelles. Have you tried something like mixing your protein into a suspension of micelles?

Be careful though, if your protein solution comes with lots of detergents; they would likely destroy the micelles.

There is a vast amount of information about reconstitution of membrane proteins into various types of lipid nanoparticles, including liposomes, nanodiscs, peptidiscs, bicelles, etc (see a few article links below for examples), but I didn't see anything about reconstituting proteins into solid lipid nanoparticles. These are usually thought of as carriers for small molecules.

Article The Peptidisc, a simple method for stabilizing membrane prot...

Article A saposin-lipoprotein nanoparticle system for membrane proteins

Article Reconstitution of Membrane Proteins into Lipid-Rich Bilayere...

https://en.wikipedia.org/wiki/Nanodisc

Article Durable vesicles for reconstitution of membrane proteins in ...