What are the differences?  I'm price neutral when it comes to these two. (I would add an extension to the Stoeger afterward)

My current lead thrower is a Winchester Defender 1300 pump which has been rock solid ... and fun to rack-BAM but it doesn't have the hold-the-trigger-and-rack-for-boom function.

The only way i would get an M3k is if I got it from MOA precision. They are not bad guns, but they take work to get running right.

the 1301 is a better gun out of the box, it is a gas gun that will shoot softer. But the earlier ones had the SN in the wrong place for opening up the loading port. the newer ones have the sn moved.

They are very different, but both work well.  The Stoeger is an inertia gun and the recoil pulse is different than the Beretta gas recoil system.  I actually have the Stoeger 20 ga 3020 set up for 3 gun and the Beretta in 12 ga.  The Stoeger will save you a few hundred bucks but it is reliable and functions well.  That being said, I love the Beretta.

I have a Beretta 1301, I barely clean the thing I shoot over a 1,000 rds. a year easily. The gun just keeps on working. Yearly I spray the bolt down with brake clean then lube with CLP. Every 3 month give it a spray of CLP.

We were at match that was really dusty, so bad handguns had to be cleaned every day. Other guys were having a bunch of shotgun problems, my Beretta never let me down including being dropped in the dirt.

I prefer gas ( in run a Versa Max CT) for the lower recoil and the 1301 is currently the fastest cycling shotgun.  The Stoeger is a good gun as well but is going to make some money and time to tune. I would go with the Beretta if between those two.