Is it possible to install a 1/2" collet on my palm router?
If I can, should I?


I'm guessing that your idea is to make a single pass with the router set to a depth of 1".  

For your project, I would not recommend the single-pass approach in MDF, or any analogous material for that matter, even with a 1/2" shank router bit with spiral flutes designed to clear the shavings out of the dado.

Instead I heartily suggest making several passes at increasing depths, removing perhaps 1/4" of depth or less on each pass.  For this, a 1/4" shank 1" diameter carbide bit would be adequate if you dropped the bit into the collet as deeply as functional, perhaps pulling it out a ways, if necessary, for the final pass.

Better yet, hog out the bulk of each of the channels in several passes with a 3/8" or 1/2" diameter router bit, then make a final pass at full depth with your (still super-sharp) 1" diameter router bit to obtain your precise width and depth.

Hope this helps.  

FWIW: I'd skip the rectangular tubing and router approach.  I'd use flat stock and cut wide kerfs that are 1" deep using a long strait-edge clamped in place and a skill saw with a heavy (thick) blade that will cut wide kerfs (e.g. 1/8" wide kerfs).

FWIW #2:  Whether you use lengths of tubing or flat stock, your door will still be prone to warping, even if you embed lengths of metal stock in multiple orientations.  To counteract the tendency to warp, you'd need to embed a rectangular metal frame.  ... or don't embed anything: use lockset hardware similar to that which is used with French doors.  That hardware will secure your door at the top and bottom, keeping it flat.

FWIW #2a: If you don't believe FWIW#2, prove it to yourself by gluing some bamboo skewers in the pattern of your choice to a sheet of heavy paper.  Once dry, you'll find that the paper is quite flexible (warp-able) even without bending the bamboo skewers.