Aside: sed expressions containing BASH variables need to be double (")-quoted for the variable to be interpreted correctly.
If you also double-quote your $BASH variable (recommended practice)
... then you can escape the variable double quotes as shown:
sed -i "s/foo/bar ""$VARIABLE""/g" <file>
I.e., replace the $VARIABLE-associated " with "".
(Simply -escaping "$VAR" as \"$VAR\" results in a "-quoted output string.)
Examples
$ VAR='apples and bananas'
$ echo $VAR
apples and bananas
$ echo "$VAR"
apples and bananas
$ printf 'I like %s!\n' $VAR
I like apples!
I like and!
I like bananas!
$ printf 'I like %s!\n' "$VAR"
I like apples and bananas!
Here, $VAR is "-quoted before piping to sed (sed is either '- or "-quoted):
$ printf 'I like %s!\n' "$VAR" | sed 's/$VAR/cherries/g'
I like apples and bananas!
$ printf 'I like %s!\n' "$VAR" | sed 's/"$VAR"/cherries/g'
I like apples and bananas!
$ printf 'I like %s!\n' "$VAR" | sed 's/$VAR/cherries/g'
I like apples and bananas!
$ printf 'I like %s!\n' "$VAR" | sed 's/""$VAR""/cherries/g'
I like apples and bananas!
$ printf 'I like %s!\n' "$VAR" | sed "s/$VAR/cherries/g"
I like cherries!
$ printf 'I like %s!\n' "$VAR" | sed "s/""$VAR""/cherries/g"
I like cherries!
Compare that to:
$ printf 'I like %s!\n' $VAR | sed "s/$VAR/cherries/g"
I like apples!
I like and!
I like bananas!
$ printf 'I like %s!\n' $VAR | sed "s/""$VAR""/cherries/g"
I like apples!
I like and!
I like bananas!
... and so on ...
Conclusion
My recommendation, as standard practice, is to
"-quote BASH variables ("$VAR")
"-quote, again, those variables (""$VAR"") if they are used in a sed expression (which itself must be "-quoted, not '-quoted)
$ VAR='apples and bananas'
$ echo "$VAR"
apples and bananas
$ printf 'I like %s!\n' "$VAR" | sed "s/""$VAR""/cherries/g"
I like cherries!