It's not clear whether you want to replace the earlier header, or add to it. Both scenarios are easy to do. Don't replace text you don't want to replace.

for file in ./*.faa;
do
    sed -i "s/^>.*/>${file%%.*}/" "$file"
done

will replace the header, but include a leading > in the replacement, effectively preserving it; and

for file in ./*.faa;
do
    sed -i "s/^>.*/&${file%%.*}/" "$file"
done

will append the file name at the end of the header (& in the replacement string evaluates to the string we are replacing, again effectively preserving it).

For another variation, try

for file in *.faa;
do
    sed -i "/^>/s/\$/ ${file%%.*}/" "$file"
done

which says on lines which match the regex ^>, replace the empty string at the end of the line $ with the file name.

Of course, your Awk script could easily be fixed, too. Standard Awk does not have an option to parallel the -i "in-place" option of sed, but you can easily use a temporary file:

for file in ./*.faa;
do
    awk '/>/{ $0 = $0 " " FILENAME);sub(/\.faa/,"")}1' "$file" >"$file.tmp" &&
    mv "$file.tmp" "$file"
done

GNU Awk also has an -i inplace extension which you could simply add to the options of your existing script if you have GNU Awk.

Since FASTA files typically contain multiple headers, adding to the header rather than replacing all headers in a file with the same string seems more useful, so I changed your Awk script to do that instead.

For what it's worth, the name of the character ^ is caret (carrot is 🥕). The character > is called greater than or right angle bracket, or right broket or sometimes just wedge.

You just need to detect the pattern to replace and use regex to implement it:

fasta_helper.sh

location=$1

for file in $location/*.faa
do
    full_filename=${file##*/}
    filename="${full_filename%.*}"
    #scape special chars
    filename=$(echo $filename | sed 's_/_\\/_g')
    echo "adding file name: $filename to: $full_filename"
    sed -i -E "s/^[^#]+/>$filename /" $location/$full_filename
done

usage:

Just pass the folder with fasta files:

bash fasta_helper.sh /foo/bar

test:

lectures

• Regex: matching up to the first occurrence of a character
• Extract filename and extension in Bash
• https://unix.stackexchange.com/questions/78625/using-sed-to-find-and-replace-complex-string-preferrably-with-regex

Locating your files

Suggesting to first identify your files with find command or ls command.

  find . -type f -name "*.faa" -printf "%f\n"

A find command to print only file with filenames extension .faa. Including sub directories to current directory.

  ls -1 "*.faa"

An ls command to print files and directories with extension .faa. In current directory.

Processing your files

Once you have the correct files list, iterate over the list and apply sed command.

  for fileName in $(find . -type f -name "*.faa" -printf "%f\n"); do
    stripedFileName=${fileName/.*/} # strip extension .faa
    sed -i "1s|\$| $stripedFileName|" "fileName" # append value of stripedFileName at end of line 1 
  done