You could try to write code (perhaps, a single regexp) which will prepend prefix to all keys in JSON string

 { "1_op": "add", "2_path": "/baz", "3_value": "qux", "4_op": "remove"
+ }
[download]

You don't even have to check if this string is a key (it's ok to damage values)
next, parse with JSON module, and when you find "1_op" and "4_op" in same hash - that would mean error.
UPD:
Alpha version:

use strict;
use warnings;
use JSON::XS;
use Data::Dumper;

my $s = <<"END";
[
{ "op": "add", "path": "/baz", "value": "qux", "op": "remove" }
]
END

my $x = 1;
$s =~ s/\"([^\"]+)\"/"\"".++$x."_".$1."\""/ge;
my $j = JSON::XS->new()->filter_json_object(sub {
my %seen;
for (keys %{shift()}) {
    die unless /^\d+\_(.*)$/;
    die "key [$1] already seen" if $seen{$1}++;
}
});
$j->decode($s);
[download]

prints

key [op] already seen
[download]

UPD2:
You can even avoid double-parsing, just remove prepended numbers in filter_json_object and return correct data. In the end yo'll get correct hash on first pass.