Since you’ve asked, “What would raising the head tube 12mm really do to the handling of the bike?” Try to understand the function of the fork in steering. The steering axis is a line through the head tube, the term “rake” is used to quantify the amount of distance the front wheel axial is set forward from the steering axis. Generally it is thought the more rake, the slower the response however the more “stable” the ride. On the other side of the geometry, the EZ rider chopper motorcycles are an extreme in steering angle; nice traveling in a straight line, but tight cornering wasn’t their forte.
I’m certain other forum members will enlighten you on additional aspects to be considered. Pokey directed you to the frame vendor, a wise consideration since raising the head tube may change the steering axis angle and in theory may affect the rake for a given fork. Be advised that with any “cycle” changing steering geometry can cause some very strange things to begin to happen at high speed, trust me on this!
Cheers; and God Bless