$str = q{TTTTTTT};
$str = s/TTT/ttt/i;
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That should be =~.

So, my question becomes, is there a way to set s/// to start looking for replacements at $-[0] + 1 when in a global context, rather than at $+[0]

In Perl 6 it will be very easy, a matter of specifying the :ov (:overlap) modifier. In 5's realm you can play with the pos function. There's been a very recent discussion about these topics.