First, I'd establish a routine when leaving the day care/preschool. We have a routine called 'going to the window'-- the preschool/day care has a window overlooking the parking lot. Whenever the boy is feeling like he's being left behind, his teacher takes him by the hand and leads him to the window. Once there, he and I both wave to each other. Sure he's a bit sad, but then he gets over it. He knows that 'going to the window' means I'm leaving.
Second, going twice a week on non-consecutive days can be really messing with his head. That's not a particularly regular schedule; he just had a day with mommy/daddy, why is this day different from the one before? He may not even know the days of the week yet, what a weekend is, or any of these other scheduling details that we as adults take for granted. I understand that there can be many considerations about why you couldn't make this a five-day-a-week event, but doing so would really stress the regularity of being dropped off.
Also, maybe hardening your heart a bit will help as well. No explanations, no cajoling, no discussion-- just say that you're dropping him off, and the drama has no effect on the outcome. There's a feedback loop at play here, where he knows he will get some dramatic attention if he has a meltdown. Don't give him the attention, and then he'll get over it. Having a leaving routine can help channel his energies into the routine rather than the meltdown, but the redirection will be helped quite a bit by knowing that the dramatic alternative is no longer available.