Hey JK,
I did a little searching and, since VC-3 was a night fighter squadron, I think those mission markings are the standard lozenge-shaped bomb insignia with a crescent moon for tail fins.
Interestingly, they look slightly different in different pictures//prints.
A painting which the pilot seems to have been awarded.
What looks to be an original era photo.
And the aircraft which seems to have seen better days.
I figured the patch was the squadron insignia, but there appear to be two(at least). One that I can’t find a time frame for.
And a more numerous one that is purportedly from the Korean War.
There’s also supposedly this patch designed by the pilot as a good candidate for the insignia. Also, the reference to dogs seems to indicate a connection to the bulldog patch above.
From this article.
Any of that help?
EDIT: If you Google “Lone Star Flight Museum Corsair” there are some in flight pics that look like a white center portion with darker areas to the left and right, which does match the color scheme of the Bulldog patch above.