Strikes me as there are 2 objectives. Restriction on max number of competitors that can represent their country and counter the limit on the number of top level competitors that can attend based on accident of birthplace.
Many other sports have a non-national grouping by invitation.
I suspect the top non-attending shooters are well known (perhaps not by all countries) so why not encourage the formation of something along the lines of "President's XI", "Barbarians XV", etc.
Open to any but selection based on verified standing over the previous 6 months (not using scores from yeas ago).
That gets the remaining top folk and their only limitation is that they cannot display a national flag on kit or apparel.
Maybe they get to become the "team to beat" :-)
BFTA Secretary 2012-2014
CSFTA Secretary 2014-2016