This seems to be hunky-dorie

But it isn't. If you start with $value being

   "<a href='foo'>bar</a>bla<br>bla<a href='bar'>foo</a>";
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you end with $value being

   "<a href='foo'>bar</a>blabla<a href='bar'>foo</a>";
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You're bitten by the notion that /PAT1'(.*?)'PAT2/ will not end up with a ' in $1.

I'd go with:

while ($value =~ /href='\K([^']*<br>[^']*)(?=')/p) {
    my ($pre, $post) = (${^PREMATCH}, ${^POSTMATCH});
    my $href = $1;
    $href =~ s/<br>//g;
    $value = "$pre$href$post";
}
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