That's the national flag of the Vatican City-State.
It's not a "Church Flag". It's a secular national emblem, same as the US Flag. As such, it is every bit as inappropriate a funeral pall as a US Flag or any other secular symbol.
A really "stylish" funeral pall (with accompanying vestments):
On the other hand, the Vatican Flag:
I, for one, don't see the resemblance.
Now, for something completely different.
You would think that if ANYBODY would have the Vatican flag on his coffin, this guy would:
But no, no Vatican flag.