Is it impossible for a retrovirus to be lysogenic?
Both things are not exclusive of each other. Retroviruses are RNA-based viruses which need to reverse transcribe their RNA genome into DNA before they can multiply. This is done by the reverse transcriptase enzyme, which is part of the virus and is incorporated into the cell.

Once the reverse transcriptase has been active and made the DNA, this DNA can also be incorporated into the hosts genome. If you look into the Wikipedia article on Bacteriophages, you will find a list of different bacteriophages, with all kind of genomes, among them as well a number of RNA-based viruses.

RNA-based viruses which enter the lysogenic pathway are obviously either pretty rare or they haven't been discovered yet in bigger numbers. Two examples are the phages F2 and MS2 (which are closely related).