How can I use a modified arrow in a tikz-cd diagram?

I need to make a commutative diagram with arrows that are themselves marked as incoming or outgoing. After some fiddling I produced this command, which produces an acceptable shape:

\newcommand{\outgoingarrow}{\ooalign{$\longrightarrow$\!\!\!\!\!\scalebox{2.4}{\raisebox{-4.5pt}{$\widehat{}$}}}}

Now I just need to use it in commutative diagrams. But it doesn't seem I can just use the shape in tikz-cd diagrams. I found this answer:

Custom arrow shapes in tikzcd

But it seems fairly cumbersome for a barely modified arrow shape like this. Isn't there some way I can get tikz-cd to simply recognize my modified arrow shape?