I guess that this is not a Perl problem. Here are some tests you should run:
Try adding explicit :80 or :443 to the URL. Doesn't work? This really seems to be a Perl problem, contact LWP support/maintainer
Try start a local TCP server on port 4430 and connect to 127.0.0.1:4430 on the ISP server. Doesn't work? Really strange, contact your ISP and ask for firewall or non-standard server configuration.
Use telnet 123.321.123.321 4430 on the ISP server and look if you get a connection (no automatic reply is expected).Doesn't work? You clearly have a network issue with your ISP or did some blocking configuration on your server.
All above worked? Try another target server and/or port for testing. I'll /msg you the URL of a private machine of mine which runs a httpd on 8080.Works? The server 123.321.123.321 doesn't like your ISP server. Get another ISP.
Still no hint? Get someone to start a webserver on port 4430 on his ISP server and try this.Works: 123.321.123.321 doesn't like your ISP server.
Doesn't work: You ISP doesn't like port 4430, get another port or ISP.

Notice:
  • All the hints are hints not this-must-be-the-reason-things.
  • There may be still some really stange OS, Perl, protocol or lib issues which would make it really really hard to find.
  • Yes, I noticed that 123.321.123.321 is not a valid IP. I just reused your placeholder for the hints. :-)

    • In reply to Re: LWP connect timeout only occurs with a port specified by Sewi
    in thread LWP connect timeout only occurs with a port specified by twotone

    Title:
    Use:  <p> text here (a paragraph) </p>
    and:  <code> code here </code>
    to format your post, it's "PerlMonks-approved HTML":


  • Posts are HTML formatted. Put <p> </p> tags around your paragraphs. Put <code> </code> tags around your code and data!
  • Titles consisting of a single word are discouraged, and in most cases are disallowed outright.
  • Read Where should I post X? if you're not absolutely sure you're posting in the right place.
  • Please read these before you post! —
  • Posts may use any of the Perl Monks Approved HTML tags:
    a, abbr, b, big, blockquote, br, caption, center, col, colgroup, dd, del, details, div, dl, dt, em, font, h1, h2, h3, h4, h5, h6, hr, i, ins, li, ol, p, pre, readmore, small, span, spoiler, strike, strong, sub, summary, sup, table, tbody, td, tfoot, th, thead, tr, tt, u, ul, wbr
  • You may need to use entities for some characters, as follows. (Exception: Within code tags, you can put the characters literally.)
            For:     Use:
    & &amp;
    < &lt;
    > &gt;
    [ &#91;
    ] &#93;
  • Link using PerlMonks shortcuts! What shortcuts can I use for linking?
  • See Writeup Formatting Tips and other pages linked from there for more info.