Is there a problem currently known to be outside class $\mathsf{NP}\cup\mathsf{coNP}$ but inside $\mathsf{BPP}$?
In your initial use of it, there was no need for "As far as we know" - We do know that

promiseBPP ⊆ promiseS2P ​ ⊆ ​ promise - (Σ2 ∩ Π2)

holds.

If ​ ZPP = NP ​ then

NP ∪ coNP ​ = ​ ZPP ∪ coZPP ​ = ​ ZPP ∪ ZPP ​ = ​ ZPP ​ ⊆ ​ BPP ​ ⊆ ​ S2P ​ ⊆ ​ Σ2 ∩ Π2

, so as far as we know

NP ∪ coNP ​ ⊆ ​ BPP ​ ⊆ ​ Σ2 ∩ Π2

could hold.

If ​ NP = coNP ​ then ​ ​ ​ BPP \ (NP ∪ coNP) ​ = ​ BPP \ S2P ​ ⊆ ​ BPP \ BPP ​ = ​ { } ​ ,

in which case there are no problems "outside class ​ NP ∪ coNP ​ but inside BPP".

In particular, no problems are "currently known to be outside class ​ NP ∪ coNP ​ but inside BPP".

I'm also not aware of any languages L known to be in BPP for which

it's also known that if L is in ​ NP ∪ coNP ​ then ​ ​ ​ BPP ​ ⊆ ​ NP ∪ coNP ​ .

"Is exactly 1 of these 2 polynomials identically zero?" ​ is "a problem currently known to be"

inside BPP but not known to be in ​ NP ∪ coNP . ​ ​ ​ More generally, non-monotone

combinations of [problems in RP that aren't known to be in coNP] yield such problems.

There is an oracle relative to which BPP is not a subset of PNP,

so presumably ​ BPP ⊆ PNP ​ is not known to hold.

If that containment is indeed not known to hold, then there pretty much can't be any known

result to the effect that "All problems in BPP are reducible to amalgamations of NP problems.".