Hello there,
This is hard to figure out from the description. I suppose if I saw you play I might have some ideas.
I cannot see how your gear would be an issue. I play mostly Strats these days, so the guitar is fine. I have not played a jazz pick in a very long time. But that is just my personal preferences. There is nothing wrong with a jazz pick. I have not tried the Dunlop Jazz III pick. I don't care for textured grips. But that pick should be fine. What thickness do you use? I prefer 1.20 mm picks, but sometimes go down to a 1.0 mm pick. I think the thicker pick has less flex and therefore I can control it better. With a thinner pick the tip is still flexing and I have already moved on. But I know guys who can trem pick using a thin pick. Besides the issue as I see it is not the trem picking technique. It is coordinating the movements of the left hand to your picking.
Been practicing for a year. Humm, I can see why you are frustrated. Speed up the metronome only when you have mastered the present speed. No set steps for moving up. As long as you need at one level. Than move up to the next for as long as it takes. My guess is at some point, you will skip along fast through different intermediate speed levels. It may take a while at one level and nearly as long at the next. But when you step up again, it may only take a few minutes to be ready to move up again. What I am saying is I would not expect you to need as much time at each speed level. More at first and much less at latter (higher speed).
Try this, it cannot hurt. Take a different practice riff to work on. For a while just use this one riff for the alternate picking practice. Do a few minutes of it maybe twice during your practice session. Take Arthur Smith's Guitar Boogie for a practice riff. That song is ideal for alternate picking. Because of the simple melody line, it works equally good at any speed. And it is a melody that will sound familiar because a lot of songs have pirated the basic melody. Use some of Guitar Boogie for your practice riff for alternate picking. Set the metronome as slow as you need to be precise, both fingering and picking. Move the speed up on the metronome only when you think you have mastered the present speed. Don't focus on the speed. Focus on precision. When you are confident you have it at the present speed, move up a notch and see how it sounds. If you find a speed level that you like the sound of, just leave it there for a while. A day, a week, however long you want. Then when you try a different speed, instead of approaching this as an exercises to increase your speed, consider it an exercises to see how the same riff sounds at other speeds. So, after you have it few levels under your belt, During your practice session, play it a say the slowest of the three you have. Then later in your practice session try it at the fastest of the three you have mastered. Then later in your practice session, try going back to the slowest level. Do your other work in between. Maybe that will work for you.
I have a hunch the problem is you are too fixated on the speed aspect and tensing up when you work on trem picking. By playing some riff slowly and working some feeling into it to see how it slows that way. Then compare how it sounds by approaching the same riff at a faster speed and maybe even a different feeling. I think by taking the focus off of "alternate picking exercises" and just use some alternate picking to see how you can shape the tone of a riff, you may get better results.
Later,