Illustrator: cannot fill transparent areas without compromising strokes using Live Paint

I've read a number of posts about this and am unsure of how to proceed.

I received an AI file that contains the following:

Here it is unselected:

However, when I group these paths/shapes and copy the element to another document, the "white" areas within the figure are not filled but are transparent.

When I try to use the Live Paint Bucket of course I CAN fill these blank spaces between the paths with white, but when I activate the tool it discards the various stroke settings which means that I lose the "fake fill" details (look at the foot to see what I mean).

Is there a simple way to address this issue? To maintain the "stroke fills" AND to fill the transparent areas with white?

Many thanks in advance!