How to change all bash variables to lowercase in sed

Question

I have hundreds of bash scripts that have upper case variables like so

#!/bin/bash

FOO=bar
RED=blue
UP=down

I am making them lower case like so

for var in FOO RED UP
do
    grep -rl '\<$var\>' | xargs sed -i 's/\<$var\>/$(echo "$var" | tr '[:upper:]' '[:lower:]')/g"
done

Now I have encountered files with hundreds of variable declerations and it is becoming too difficult to copy paste each one. I tried automating the process like so

grep -rl '\w*=[^=]' | xargs sed -E "s/(\w*)=([^=])/$(echo \1 | tr '[:upper:]' '[:lower:]')=\2/g"

But the problem lies in $(echo \1...) which obviously will fail

Is there a way to quickly find all vars and convert to lowercase using regex?

EDIT:

As requested, assume we have 1 file like so

#!/bin/bash

AAA="hello world"
AAB="hello world"
...
ZZZ="hello world"

And assume I have a large number (>10000) nested (in directories or sub-directories) like so

#!/bin/bash

echo $AAA
echo $AAB
...
echo $ZZZ
if [ $AAA == "foo" ]
then
    #do something
fi

I want to search for any assignments ( \w*=[!=] ) and then do a word replace on that variable to make it lower case. Note the if condition, I through that in there to show that there may be a case of a word followed by 2 equations, but we are matching for only 1 equation (and no space)

Edit 2: Note that I want only the variable names to be converted to lower case, not everything

# Current Input
SOME_VAR="Humpty Dumpty had a great FALL"

# Desired Output
some_var="Humpty Dumpty had a great FALL"

Answer 1

The only solution that worked for me was to go through and replace the variable names to lower case in three steps

 # Change all SOME_VAR=... to some_var=...                                 
 grep -rl '\w\+=' | xargs sed -i 's/\w\+=/\L&/g'                             
                                                                             
 # Change all $SOME_VAR to $some_var                                         
 grep -rl '$\w\+' | xargs sed -i 's/$\w\+/\L&/g'                             
                                                                             
 # Change all ${SOME_VAR} to ${some_var}                                     
 grep -rl '$\w\+' | xargs sed -i 's/${\w\+}/\L&/g'

Then I went back and changed the few environment variables back to upper case