How it goes if you set user agent's ssl_opts?

$ua->ssl_opts( SSL_version => 'TLSv1');
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LWP's ssl_opts description says

Other options can be set and are processed directly by the SSL Socket implementation in use. See IO::Socket::SSL or Net::SSL for details.

So I guess if you have some luck with IO::Socket::SSL, opts will do something. But I am not sure.