Mmm. dunno. I've already spent far too much time researching your problem, when I have real work to do! :). I would guess no, it's not possible for the same script to, since it looks like http10.pm will only make HTTP/1.0 requests. Perhaps set/unset the ENV variable and spawn off scripts that make the requests?. Good luck!
Update
Perhaps AM's suggestion is a direction to try... here's some sort-of tested code to play with.
use LWP::UserAgent;
require LWP::Protocol::http10;
my $url = "http://localhost/";
my $ua = LWP::UserAgent->new();
my $req = HTTP::Request->new(GET => $url);
# make HTTP/1.1 request
LWP::Protocol::implementor('http', 'LWP::Protocol::http');
$ua->request($req);
# make HTTP/1.0 request
LWP::Protocol::implementor('http', 'LWP::Protocol::http10');
$ua->request($req);
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