\bigoplus, \prod, etc. with hats

Question

I would like to have big operators like \bigoplus, \prod, etc. with hats. A first approximation is the following code:

\documentclass{scrartcl}
\usepackage{unicode-math}
\DeclareMathOperator{\compdirsum}{\widehat{\bigoplus}}
\DeclareMathOperator{\restprod}{\widehat{\prod}}
\begin{document}
(
\displaystyle
\compdirsum_{i \in I} V_i \to \restprod_{i \in I} V_i
)
\bigskip
(
\compdirsum_{i \in I} V_i \to \restprod_{i \in I} V_i
)
\end{document}

However the hat is not centered and not «wide»:

How can this be fixed?