Edit

My previous response was wrong -- I had not read your question correctly. You want the complete request.

  DB<3> p $req->as_string()
GET http://search.cpan.org/dist/libwww%E2%88%92perl/
Accept: text/html
[download]

nope..

 p $req->dump
GET http://search.cpan.org/dist/libwww%E2%88%92perl/
Accept: text/html

(no content)
[download]

Not there either...

If you debug waaaay down into the code you'll find it in an object stored in a variable but nothing you can access externally:

LWP::Protocol::http::request(/usr/lib/perl5/site_perl/5.10.0/LWP/Proto
+col/http.pm:226):
226:        if (!$has_content || $write_wait || $has_content > 8*1024)
+ {
  DB<17> p $req_buf
GET /dist/libwww%E2%88%92perl/ HTTP/1.1
TE: deflate,gzip;q=0.3
Connection: TE, close
Accept: text/html
Host: search.cpan.org
User-Agent: lwpcooktest.pl/0.1 libwww-perl/6.03
[download]

The previous response about capturing the request with an echo server of some kind is the safest way to go.