So, the way I've read and understand the Arsenal Chaplain's ability (because I did a similar trick) is that the warpriest doesn't get multiple weapons like a fighter does unless he takes weapon focus to make a weapon a sacred weapon as the weapon training applies to all a warpriest sacred weapons.
This is important because a warpriest never gets additional weapon training groups to normally qualify for Advanced Weapon Training that way, however the Advanced Weapon Training feat can be taken once every 5 levels and only requires you to be a 5th level fighter with weapon training. Warpriest get to count as fighters for selecting feats, so you can take the feat to pick up Advanced Weapon Training abilities.
Now reading through the Fighter VMC and re-reading through Advanced Weapon Training, this will work but not exactly like you're reading it. Advanced Weapon training specifies "Beginning at 9th level, instead of selecting an additional fighter weapon group, a fighter can choose an advanced weapon training option for one fighter weapon group that he previously selected with the weapon training class feature"
At 11th level you don't actually have any fighter weapon groups, you have some sacred weapons that benefit from Weapon Training but no actual weapon groups. So at 11th level you would have to choose a weapon group (which wont stack with the bonus your sacred weapons are already getting) and couldn't choose to replace with with an advanced weapon training. Now when you get to 19th level and get weapon training 2 you can trade out the second group for an advanced weapon training.
Overall, this is a terrible way to do it and you're much better doing it the way I suggested and just using your feats to select Advanced Weapon Training feat once every 5 levels.
In regards to Sash of the War Champion or Gloves of Dueling, it has been stated (by the person who wrote Advanced Weapon Training IIRC) that he intended for those items to count into any of the effects of the Advanced Weapon training.
They stack according to this faq.
Gloves of duelling add +2 to your weapon trianing bonus, as normal.
You may take the Advanced Weapon Training Feat, but you need to count as a 5th level Fighter for the purpose of selecting said feat.
They do according to this faq
FAQs can set precedent, but I don't think this one applies in this case. The Myrmidarch Magus gets weapon training that scales at the same rate (or not as good) as a fighter does. And that FAQ is about stacking fighter levels with magus Myrmidarch levels.
If an Arsenal Chaplain Warpriest was multiclassing into fighter I'd be fine saying the FAQ applied, since the Arsenal Chaplain also gains Weapon Training at the same rate a fighter does.
But were talking about Variant Multiclassing instead of actual multiclassing. If you allow them to stack, the Arsenal Chaplain VMC fighter would have a higher Weapon Training Value than a pure fighter. Which definitely isn't right.
Ergo, I'm fairly certain that FAQ doesn't apply because it's about multiclassing and not VMC.
It doesnt apply because it's about multiclassing and not Variant Multiclassing?
I agree it might not work because the Chaplans weapon training works differently than the Fighters - they might share the same name, but they are not fuctionally the same ability - but there is no reason why VMC fighter should interact with a Sohei or a Myrmidarch any differently than the FAQ.
Other than "I don't like it" or "I think it's overpowered."
It doesnt apply because it's about multiclassing and not Variant Multiclassing?
Yes, they work in vastly different ways.
Because you're always taking levels in the base class you're always progressing your Weapon Training from the base class. An Arsenal Chaplain Warpriest gets the same progression to his sacred weapon weapon training as a fighter gets with his first weapon group. But if you let VMC fighter weapon training stack with the warpriest weapon training such a character would end up with a higher bonus from Weapon Training than a pure fighter could get. How in the world does that make any sense? It doesn't. So I have to say that FAQ doesn't apply.
edited my earlier post - you responded just as my edit was completed
...And if you wear Gloves of Dueling you add another +2 on top of that, maybe we should ban those too.