I need to install a new header to enlarge a window
(in a load bearing wall).
avoid the problems of shrinkage.
only green lumber is available ... for anything larger than 2 x 4's.

Around here (Pacific NW) I can get kiln dried up to 2x8's or so, but 2x6 is
about the largest size that's easy to find.

In any case, I used green 2x10's for our door and window headers and
haven't noticed any shrinkage problems (no drywall cracks).

use 2x (insert width here) kiln dried stock and nail them together
using 1/2" plywood in between to fill the width.

That assumes a 2x4 wall. In our house (2x6 walls), I used two 2x10's, one
on the outside of the wall, and one on the inside of the wall. Then I
filled the space in between with fiberglass insulation. Our largest span
was 5' and only had the roof load above, so this worked very well. Of
course, the success depends on the span and the load it is carrying.

buy the thick stock beam you need

A solid beam weighs more, will probably shrink more, and doesn't provide
much insulating value. Also, since this sounds like a remodeling situation,
I would think it would be harder to get a solid beam in place than a couple
of 2x's.

Anthony