The easiest way to deal with self-signed certificates is to expect exactly this certificate. This can be done with the SSL_fingerprint argument of IO::Socket::SSL which you can set using ssl_opts in LWP. With this argument all other checks (subject matches hostname, expiration...) are disabled.
You need at least IO::Socket::SSL version 1.980 for this (released 04/2014). See the documentation of SSL_fingerprint for details but in short:

To get the current fingerprint analyze the servers certificate or get the fingerprint from a trusted location with:

use IO::Socket::SSL;
my $cl = IO::Socket::SSL->new(
    PeerAddr => "ip:port", 
    SSL_verify_mode => 0
);
print $cl->get_fingerprint."\n"; # sha256$b64d26746ca1cd5403....
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And to use it:

my $ua = LWP::UserAgent->new...
$ua->ssl_opts(SSL_fingerprint => 'sha256$b64...');
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