I came upon a page that talks about tricking sed into doing what I want, but the problem is that I don't understand the document. It's too advanced (for me).
https://learnbyexample.github.io/sed-lookarounds/

Anyway, I finally found a solution, and it was with the help of AI... which of course probably just found this line from some website:

echo "$NUM" | sed -r ':a;s/^([-+]?[0-9]+)([0-9]{3})/\1,\2/;ta'
[download]

And I wrote the following enclosure:

# This function inserts commas into numbers at every 3 digits
# and returns the result in a global variable called $STR.
#
function Commify
{
  STR="$1"
  if [[ "$STR" =~ ([\+\-\$\(0-9\.]+) ]]; then
    local NEG=''
    local NUM="${BASH_REMATCH[1]}"
    local PREFIX="${STR%$NUM*}"
    local SUFFIX="${STR##*$NUM}"
    if [[ "$NUM" == *"+"* ]]; then NEG="+"; fi
    if [[ "$NUM" == *"-"* ]]; then NEG="-"; fi
    if [[ "$NUM" =~ 0+([1-9]+[0-9.]*) ]]; then NUM=${BASH_REMATCH[1]};
+ fi
    NUM="$NEG$NUM"
    NUM=$(echo "$NUM" | sed -r ':a;s/^([-+]?[0-9]+)([0-9]{3})/\1,\2/;t
+a')
    STR="$PREFIX$NUM$SUFFIX"
  fi
}
[download]

I could possibly put this in a while loop and make it commify every number in a string, but I think this is fine as it is. Let's not overcomplicate things too much... Btw I can't help but wonder how amazingly similar Bash scripting is to Perl. There are soo many similarities!