Or is it as simple as writing \x{efbbbf} as the first thing after the HTTP headers?

The string my $str = "\x{efbbbf}"; does not contain the BOM character, it contains U+EFBBBF, which is not valid a valid Unicode character (AFAIK: I believe Unicode only goes to U+1FFFFF U+10FFFF). The string my $str = "\x{feff}"; contains the BOM character.

If you did use the string you suggested, whether with raw mode or with UTF-8 output encoding, you will not get what you thought:

C:\Users\Peter> perl -e "binmode STDOUT, ':raw'; print qq(\x{efbbbf})"
+ | xxd
Wide character in print at -e line 1.
00000000: f8bb bbae bf                             .....


C:\Users\Peter> perl -e "use open ':std' => ':encoding(UTF-8)'; print 
+qq(\x{efbbbf})" | xxd
Code point 0xEFBBBF is not Unicode, may not be portable in print at -e
+ line 1.
00000000: 5c78 7b45 4642 4242 467d                 \x{EFBBBF}
[download]

Neither of those outputs the UTF-8 bytes for the BOM U+FEFF character.

Instead, you either need to manually send the three octets separately in raw mode, or use raw mode and manually encode from a perl string into UTF-8 bytes, or use UTF-8 output encoding and send the U+FEFF character from the string directly:

C:\Users\Peter> perl -e "binmode STDOUT, ':raw'; print qq(\xef\xbb\xbf
+)" | xxd
00000000: efbb bf                                  ...

C:\Users\Peter> perl -MEncode -e "binmode STDOUT, ':raw'; print Encode
+::encode('UTF-8', qq(\x{feff}));" | xxd
00000000: efbb bf                                  ...

C:\Users\Peter> perl -e "use open ':std' => ':encoding(UTF-8)'; print 
+qq(\x{feff})" | xxd
00000000: efbb bf                                  ...
[download]

Whether or not that would "work" in your use-case is something I don't know: my guess is that it won't help, because anything that's using HTTP headers should be paying attention to the encoding listed in the headers, and not requiring a BOM in the message body. Though I guess if it's saving the HTTP message body into a file, and then later using that file, maybe the BOM would help. I don't know on that, sorry.

--
warning: Windows quoting used in code blocks; swap quotes around if you're on linux