All birdshot should fire fine.
Question 1) - How many rounds are through your gun? 200+?
Question 2) - Is your bolt carrier drenched in lube? Like literally, drenched.
I've come to believe that failure to reliably cycle birdshot comes from one of these two factors, each and every time it comes up on these forums.
My first M4 wouldn't cycle anything reliably for the first ~150 rounds. Buckshot, slugs, birdshot ... it would fire 3-4 rounds and have various jams on the next. It didn't have a drop of lube on it, as I was a newb and I didn't figure it would matter.
12000 rounds later now and I've yet to have a single malfunction after I lubed up the bolt carrier. I've cleaned it probably 4 times in all those rounds.
Between 3 friends and myself, after my initial "dry" m4 fail prone firing, we have purchased and broken in 4 M4's between us. Each of them has been 100% flawless out of the box with every type of ammo, because before initial firing we slathered machine gunners lube or another dedicated high temperature gun oil on the bolt carrier and recoil spring.
Member "Unobtanium" on this board has been through 3, maybe 4 M4's by now, and each has been 100% out of the box as well because he knew to properly lube.
As such, every thread I see that related to this topic I've somewhat concluded boils down to these two things, plentiful lube and 200+ round count to make sure an overly tight tolerance isn't causing friction based stoppages (unless something is catastrophically wrong, which I have yet to see).
So I ask, how many rounds are through your gun? And is the interior recoil spring, bottom of the bolt carrier and receiver rails slathered up with oil AT THE TIME OF THE SHOOT? Cleaning and oiling with CLP a week prior to shooting often leads to a near completely dry gun at the time of the shoot ... as many multi purpose or cheaper oils evaporate at room temperatures or as soon as they start to heat up. Oil right before your shoot, proper lubrication is 100000x more important than cleaning when it comes to this gun (and most other guns, for the matter).
If you have 200+ rounds through your gun, and you're lubed up, you should be able to burn through any brand of standard birdshot rounds without any issues. If not, detail the issue here and see if we can diagnose it.