A stainless steel solder paste stencil, which now costs about $25[1], would probably work well. Those are thin (3 to 8 mils), so the parts aren't going to get stuck in a deep hole. They're usually used in a hinged frame that puts them under tension and makes them rigid. Use one stencil for solder paste, then change to the placement stencil for part placement.

Stencils used to cost about $200, but at $25, this starts to look like a viable option.

[1] https://www.oshstencils.com/#%20