So.... it's complicated. It looks like the stat weights are real, and they're a weird interaction between the T15 4-piece
and Divine Purpose
and your current stats.
Basically because of the 4-piece and Unbreakable Spirit, you're getting 25% uptime on Divine Protection (10 sec. duration, recast every ~40 sec). That means 25% of the time you're being rewarded (with more Holy Power) for taking extra damage.
It's not related to Unbreakable Spirit itself though - removing that drops you to 17% uptime, but you still get the funky stat weights.

However, if you swap your
Focused Shield glyph to
Divine Protection, all of the sudden your stat weights switch back to what we'd expect:

Similarly if we arbitrarily add 1000 stamina to your gear via the tabard slot:

To see why, let's take a look at several of the stats that differ between the three situations, plus another two where I add 250 or 500 stam. Note that these aren't cumulative, they're all relative to the Base configuration.
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DP uptime SotR uptime HP_T15_4pc Base TMI Stat Weights
Base 25.82% 86.50% 136.07 611.6 Funky (Sta/Armor negative)
+250 stam 25.81% 86.54% 136.00 574.2 Funky (Sta/Armor negative)
+500 stam 25.46% 83.22% 134.26 819.5 Normal (Hit>Exp>Sta>Mast>Armor>Str>Haste)
+1k stam 25.08% 83.18% 134.24 713.2 Normal (Hit>Exp>Sta>Mast>Armor>Str>Haste)
FS->DP 24.59% 74.50% 129.50 676.4 Normal? (Hit>Exp>Haste>Sta>Str>Mast>Armor)
The HP overflow is much lower in the FS->DP sim, but it's fairly similar in the other two so that's probably not a major factor.
I think what we're seeing here is that you're near a
Stamina breakpoint, believe it or not. Adding 250 stamina improves your TMI without having a significant effect on DP uptime, SotR uptime, or the Holy Power income from 4T15. However if we bump that up to 500 stam, all of the sudden your T15 HP income drops by 2 holy power, your SotR uptime suffers by 3% as a result, and your TMI actually goes
up!. Moving further to 1000 stamina starts to reduce your TMI again as it should, again.
Basically, what's happening is that you're crossing an HP threshold at which some type of attack is no longer doing enough damage to break the T15 4-piece Holy Power threshold, and thus reducing your HPG. Remember that the sim determines your scale factors by adding 1000 of each stat, so when it does this it sees your TMI go up and calculates the stat weight accordingly.
That explains why the FS->DP glyph change gives the same effect - it's reducing the damage you take during DP by 20% now, so it has an even larger effect on your T15 4-piece HP income. It also explains why you see a similar effect when adding armor (same basic idea - reducing damage intake relative to health and crossing the same threshold).
And finally, it also explains why the stat weights flip. Once you've crossed that threshold, your stat weights return to normal because adding more stamina improves your TMI again. It's only near that threshold where things get weird.
This highlights one of the issues I've always had with the T15 4-piece bonus. It can have strange negative interaction with other stats that improve your survivability. While dodge/parry/str. don't go negative, I'm willing to bet that the 4-piece is reducing their effectiveness (though that's tougher to test empirically).
That interaction isn't generally a problem when the effect is weak, like the T16 2-piece, which has the same problem, but probably isn't strong enough to exhibit this sort of influence. But the T16 4-piece effect isn't always weak - it becomes a very strong effect when you're near one of the stamina thresholds, where you go from generating 0 to 1, or 1 to 2 Holy Power from a given attack.
Anyway, I'm glad I got to the bottom of it, and this is fascinating stuff. It'll make a great blog post.
Unfortunately, that doesn't help your situation. But maybe this will. The GUI doesn't have a built-in option to change the amount of each stat it uses to calculate the stat weights (mental note: add this). However, if you add the line
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scale_delta_multiplier=0.2
in the simulate window (I put it right after your spec=protection declaration), it will use 0.2*1000=200 of each stat to calculate the scale factors. This is below your threshold crossover point, so you end up getting normal-ish stat weights (though the error is a little larger, probably because we're approaching the breakpoint).
The other option is to realize that you're very near the breakpoint and shouldn't be adding much more stamina. We know it's between 250 and 500 stamina above your current gear. If you want to find it more precisely, you can change the line containing your tabard to:
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tabard=renowned_guild_tabard,stats=350sta
And tweak the stamina value little by little until you get normal stat weights again. If you want to do this faster, you can skip the stat weight calculation and just look for a sharp increase in TMI, which should tell you that you've crossed the threshold. That way you know that you shouldn't add any more than that much stamina.
(By the way, testing with 350 sta gave me a TMI of 827.4 and normal stat weights, so your target range is somewhere between 250 and 350 stamina from your current setup.
Hope that was helpful....