Yes, you can use "since" instead of "because" in any tense. But the most natural way to say it would be either:
1) Since you had eaten all the desserts, we couldn't eat any
2) Since you ate all the desserts, we can't eat any dessert
The first is exactly what you posted in your question. A rephrasing illustrates why it's correct: "We could not eat any desert yesterday because you had eaten it all the day before." "You" ate the deserts at some time prior to the event reported in the current sentence, so the past perfect is used to report "your" actions, and the simple past to report ours.
The second sentence here can be rephrased like this: "We can not eat any desert now because you ate it all yesterday." We are reporting a current situation which was caused by "your" actions in the past.
As for your second example:
Since you ate all the desserts, we couldn't eat any dessert.
It doesn't seem so much grammatically wrong as logically awkward. "Your" actions here must have happened before "ours"--so they must have occurred at some point in the prior past. Thus the natural assumption here should be that the past perfect ought to be used. But if somehow the context of the sentence implies that both happened simultaneously at some time in the past, they could both be in the simple past tense. However, this seems like an unlikely scenario, and the construction would probably still seem awkward even if there was some plausible situation to justify it.