• I couldn't get interpolation of constants declared using (pun intended) the constant module and then ...
• I read PBP - which denegrates the use (again, pun intended) of the constant module

In retrospect, it occurs to me that I used the latter as a convenient excuse/motive for giving up on attempts to achieve the former.

IIRC, PBP deprecates the use of constant in favor of the Readonly module, which gives you (among other things) read-only scalars which interpolate as expected in double-quotish things like strings and regexes.

I can't test this right now, but I think the following works and also gives you a more elegant regex interpolation:

use warnings;
use strict;

use Readonly;

Readonly my $BAR => qr{ bar }xms;

my $string = q{fie foo bar baz barf};

print qq{found $1} if $string =~ m{ foo \s* ($BAR) \s* baz }xms;
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Should print:

found bar
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You're not wrong (about the deprecation in favour of Readonly) ... and yes, I wholeheartedly agree with you WRT the elegance of the result.
A user level that continues to overstate my experience :-))