These are the settings that work for me, as of this posting. I use BOTH Dubs Apparel Tweaks AND KV Show Hair With Hats. Just one or the other doesn't seem to work correctly, though I haven't tested this extensively. Bear in mind that my extensive mod list includes abominations such as NALS Facial Animation and Nyaron Race Lite, so YMMV with your modlist.

If anyone finds any mod/option combinations that produce better results, please post here instead of getting buried in the comment section.

Option 1:
Pros: This results in hair showing under hats, hats only showing while drafted, no hat/hair flickering.
Cons: This results in nonhuman races having hair rendered twice. This is due to nonstandard head sizes of races like the Nyaron.

In the mod menu, under Dubs Apparel Tweaks, I have the following options selected:
[X] Show Apparel Only When Drafted
[X] Show Hair Under Hats
Hair Layer Offset -.002 (you'll have to carefully drag the slider)
Hat Scaling 1.1 (you could probably get away with shrinking this to 1.0)

In the mod menu, under Show Hair, I have the following options selected:
[X] Show Hats Only When Drafted
Hats: [Set All] is set to "Show Hair Only Drafted HH SB"

Option 2
Pros: Works with all races/head sizes. Hair draws as normal, helmet draws as normal.
Cons: Hair is not shown under hats.

In the mod menu, under Dubs Apparel Tweaks, I have the following options selected:
[X] Show Apparel Only When Drafted

In the mod menu, under Show Hair, I have the following options selected:
[X] Show Hats Only When Drafted
Hats: [Set All] is set to "Show Hair Only Drafted HH SB"