As long as the braces groups are not nested you could do it by separating blocks in a split and handling them differently.

use Data::Dump;

my $data = q|
foo
bar
{{
alpha
beta
}}
baz
|;

@splits = split /({{.*?}})/s, $data;

dd \@splits;

my $result="";
while (my $block = shift @splits) {
  $block  =~ s/\n/<br>\n/gs;
  $result .= $block;
  $result .= shift @splits if @splits;
}

print $result;
[download]

output

["\nfoo\nbar\n", "{{\nalpha\nbeta\n}}", "\nbaz\n"]
<br>
foo<br>
bar<br>
{{
alpha
beta
}}<br>
baz<br>
[download]

I refrain from trying a complicated and potentially unmaintainable one-line regex solution.

Some come to mind¹, but I don't see the necessity if there are no other restrictions (like lack of memory) involved.

UPDATE

¹) like

• looping with while (/({{.*?}})/gs) (and \G and pos)
• using /e to do substitution within substitutions
• complicated look-ahead and look-behind assertion
• using \K somehow to restrict replacement