How many operad structures are there on the symmetric sequence of simplices / finitely-supported probability measures?

Put $Q_n=\{(x\in [0,1]^n:\text{max}(x_1,\dotsc,x_n)=1\}$. Then there is an isomorphism $Q\to P$ of symmetric sequences given by $x\mapsto x/\sum_ix_i$. Define $$ p \circ (p_1,\dots,p_n) = (\min(p^1,p_1^1), \min(p^1,p_1^2),\dots, \min(p^1 ,p_1^{n_1}),\min(p_2,p_2^1), \dots,\min(p^n,p_n^1), \dots,\min(p^n,p_n^{n_n})). $$ This gives an operad structure on $Q$. Let $C$ be the commutative operad, so that $C_n$ is a singleton for all $n$. There is an operad morphism $C\to Q$ sending the unique point of $C_n$ to $(1,\dotsc,1)$, but there is no operad morphism $C\to (P,\circ^f)$, so $Q\not\simeq(P,\circ^f)$ as operads.

Abusing notation, write m for the uniform distribution on each finite set and define p o p' = m for any p, p' not equal to the identity.

This is possibly a limit of operads of the formula you give where you choose homeomorphisms which tend (pointwise) to a constant function f(x) = c. But I think qualifies as an operad structure not in your suggested classification.