Consider the category whose objects are non-negative integers that are represented as dots along a line, and whose morphisms are generated by $X$---positive crossing, $\bar{X}$ --- negative crossing, $\cup$, $\cap$, $Y$ --- co-multiplication, $\Lambda$ --- multiplication, and of course, $|$--- the identity morphism. The relations in the category are the usual braid relations, adjointness for $\cup$ and $\cap$, distant $X$s and $Y$s commute, associativity, co-associativity, $(Y \otimes |) \circ X = (| \otimes X) \circ (X \otimes |) \circ (| \otimes Y)$ and variations, and relations involving $\cap$ and $Y$ to relate this to $\Lambda$. In brief, the relations in $\cup$, $\cap$, $Y$, and $\Lambda$ satisfy the Frobenius algebra axioms; the relations for $X$ satisfy the braid relations, and the obvious relations for crossings and trivalent vertices hold.
Is this structure is a free braided Frobenius category? I might not have the adjectives in the correct order. It seems that the structure should be the most free that satisfies braiding, Frobenius, and the intermingling of the two structures.