Having completed in the first three years all the courses needed, dedicated my last year of studies at/in the Digital Communications Laboratory.
I think it is more correct to use in because Laboratory is something really specifi c
I dedicated my last year of studies to working in the digital communications laboratory.
As for "in" and "at", here's how they work.
Where's Joe? He's in the laboratory. - The speakers are in a building that has a laboratory. Use "in" when the place is close and easily accessible. As well, use "in" for places we understand are enclosed spaces.
Where's Joe? He's at the laboratoy. - The speakers are not in the building in which the laboratory is located. They would probably have to drive to the laboratory, use public transportation, or walk for at least a few minutes. Use "at" when speaking about a remote location.