Replacing spaces with underscores within each capture group in Vim using regex

Question

I often edit Power Query "M" code using Vim while working with Power BI. Typically I prefer to change the auto-generated identifier names for each query step by replacing spaces with underscores and shifting it to lowercase. Identifiers with spaces are represented as #-prefixed quoted strings such as #"Change Column Types". For that example, I would like every instance to be transformed to change_column_types instead. I would like to create a keymapping that can do this within any buffer for all instances.

An example file is shown below, followed by the desired 'cleaned' file:

Input:

let
    Source = Excel.CurrentWorkbook(){[Name="Table1"]}[Content],
    #"Changed Type" = Table.TransformColumnTypes(Source,{{"id", Int64.Type}, {"foo", type text}, {"bar", type text}, {"baz", Int64.Type}}),
    #"Filtered Rows" = Table.SelectRows(#"Changed Type", each ([foo] = "sdf")),
    #"Grouped Rows" = Table.Group(#"Filtered Rows", {"bar"}, {{"Count", each Table.RowCount(_), Int64.Type}})
in
    #"Grouped Rows"

Desired output:

let
    Source = Excel.CurrentWorkbook(){[Name="Table1"]}[Content],
    changed_type = Table.TransformColumnTypes(Source,{{"id", Int64.Type}, {"foo", type text}, {"bar", type text}, {"baz", Int64.Type}}),
    filtered_rows = Table.SelectRows(changed_type, each ([foo] = "sdf")),
    grouped_rows = Table.Group(filtered_rows, {"bar"}, {{"Count", each Table.RowCount(_), Int64.Type}})
in
    grouped_rows

The trick is that this involves trying to replace multiple (an unknown number of) characters within each capture group. Since there are only usually 2 or 3 words in each, I can handle these in a hacky way using two ex commands:

:%s/\v\#"(\w+)\s(\w+)"/\L\1_\L\2/gc
:%s/\v\#"(\w+)\s(\w+)\s(\w+)"/\L\1_\L\2_\L\3/gc

However, this obviously does not seem ideal given that it is hardcoded to support only certain numbers of words. I have tried doing some nested grouping but the problem still seems to be the same. Is there a way to define a replacement pattern within some other replacement operation? Any help on how to properly handle this would be appreciated.