On the other hand it's in field where I can't really help us out since my best idea for dealing with the Faith is and will remain to call out the Father to settle this one on one, Avatar vs Dragon and the winner gets Westeros.
Not a joke by the way.
Let's look at what we and the Seven offer in comparison?
Father: Justice and good lordship.
Our Justice system, far from perfect as it is, is fairer than anything Westeros has, which the Faith approves of so far. And our lands are blooming, with more and more immigrants daily.
Warrior: Security from outer and inner threats.
We are a hell of a lot better at dealing with both obvious demonic and Deep threats than Westeros has proven so far, we are also less riddled with Devil cults and other infiltrating foes. Not a difficult bar to clear, given that they let a Mindflayer buy large amounts of Wildfire in KL.
Mother & Maiden: Charity, healing, mercy.
We have a solid social security, better healthcare than any non-High-Magic nation could possibly achieve and a punishment system that activly works to reform people, rather than cutting of limbs and letting them go on as before.
Smith: Craft and economy.
Not only are we constantly building things that would have counted as Worldwonders a few years ago, our economy is also constantly growing at an insane rate (I'm assuming that still happens for now, we'll see how it goes without Azel)
Oh, and we have many dozens of magecrafters improving our Imperium and are slowly working our way towards a magictech industrial revolution.
Crone: Knowledge and Wisdom
We are not only gathering incredible amounts of knowledge, we are also efficiently working on spreading to all people through public education, the Scholarium and recently a new university.
Stranger: Well, who really needs humility before death anyway?
So all things seen from above we are objectivly doing a better job than the Seven in all things their devout followers should care about.
Therefore I can only conclude that a demonstration that they have no divine wrath to fear for we are more powerful in this realm than a god sending down what power he can would be the capstone that could convice every rational man or woman in Westeros that they should choose us over the Seven.
Edit: And for the afterlife-worries we can always offer the talk to an actual Angel how he got to heaven long before the Seven were a thing.