Clean, 238 bytes

import StdEnv,Data.List
$v m#p=[[(x,y)]\\x<-[0..]&l<-m,y<-[0..]&' '<-l]
=or[and(zipWith(>=)(s++[0,0..])v)\\s<-permutations(map length(foldl(\p _=nub[sort(nub(e++[(x,y)\\(u,v)<-e,x<-[u-1,u,u+1],y<-[v-1,v,v+1]|(m!!x)!!y<'C']))\\e<-p])p p))]

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Defines the function $ :: [Int] [[Char]] -> Bool. TIO link includes a wrapper around STDIN.

Expanded:

$ v m // in function $ of v and m
    # p // define p as
        = [ // a list of
            [(x, y)]    // lists of pairs (x, y)
        \\              // for each
            x <- [0..]  // index x
            & l <- m    // at list l in m
        ,               // for each
            y <- [0..]  // index y
            & ' ' <- l  // at spaces in l
        ]
    = or [ // true if any of the list of
        and (               // true if all of
            zipWith         // element-wise application of
                (>=)            // greater than or equal to
                (s ++ [0, 0..]) // permutation s padded with zeroes
                v               // list v of volumes
        )
    \\                      // for each
        s <- permutations ( // permutation s of
            map length (    // the lengths of
                foldl       // a left-fold of
                    (\p _   // function on p discarding second argument
                        = nub [ // the unique elements of the list of
                            sort (          // sorted versions of
                                nub (       // unique lists composed of
                                    e       // list e of points in a region
                                    ++ [    // prepended to the list of
                                        (x, y)      // pairs (x, y)
                                    \\              // for each
                                        (u, v) <- e // pair (u, v) in list e
                                    ,               // for each
                                        x <- [u-1, u, u+1] // x-coordinate adjacent to u
                                    ,               // for each
                                        y <- [v-1, v, v+1] // y-coordinate adjacent to v
                                    |               // where
                                        (m!!x)!!y < 'C' // the character in m at (x, y) is a space
                                    ]
                                )
                            )
                        \\          // for each
                            e <- p  // region e in p
                        ]
                    )
                    p // applied to a starting argument of p
                    p // recursively, for each element in p
            )
        )
    ]