This is a short question:
In the symmetric monoidal category of chain complexes (over a field if necessary), does the endomorphism operad carries a Hopf structure, i.o.w. can it be considered as a Hopf operad?
The answer is probably no, but just to be sure
No. Given any Hopf operad $\mathcal{O}$, its arity one part $\mathcal{O}(1)$ is a bialgebra. In particular, you would get that $\mathrm{End}_V(1) = \mathrm{End}(V)$ is a bialgebra.
Suppose that for a vector space $V$, $\mathrm{End}(V)$ is a bialgebra. Then the counit is an algebra map $\epsilon\colon \mathrm{End}(V)\rightarrow k$. The algebra $\mathrm{End}(V)$ is simple, so $\epsilon$ is injective which can only happen if $V$ is a line.
