Look at it this way...
The BATF will issue you a Form6NIA when you use a hunting Lic. because with a hunting lic. you fall within one of the "exceptions" to the prohibition on NIA's possessing/importing firearms into the USA... That's it. That's all the BATF cares about... They're following the letter of the law.
When you roll up to US Customs/Immagration and they ask where you're going or what the purpose of your trip is and you say...
"I'm going to a shooting match at ###"
They might, in addition to your Form6NIA ask for something like a match invite to support your
stated reason for entering the USA on
that occaision.
The same can and has happened if you were to be traveling to Chicago for a buddies wedding...
"Where are you going?"
"To Chicago for a friends wedding"
"Do you have an invitation?"
"Why... yes I do"
"Let me see it"
It's like when you say you're going to Fla. to visit grandma and they ask you exactly where grandma lives... They're just trying to see if you bull####ting

I don't get all sweaty over it... I usually have something with me showing a date/time for a match I'm attending... I've used everything from invites printed on club letterhead to simple copies of confirmation emails as well as screen captures of a website showing the date/place for an event...
'Never been hassled.