memes are their own genes

That is the intended idea, yes. "Genetic" - under such usage would mean - "to do with heritable information" - and NOT the nucleic-acid centric "to do with heritable information stored in DNA".

This would make all transmissible human culture "genetic". If memes are a type of gene which is not made of DNA, then that throws quite a spanner into the terminology of many "genes" vs "environment" debates.

But their genes can't reasonably be treated as our genes. The life-cycle is wrong.

Indeed - though that is actually the position of major theorists in the area, Boyd and Richerson.

They think that culture is part of the human extended genotype - despite, as you say, their life-cycles being totally different.

In: "Culture is Part of Human Biology Why the Superorganic Concept Serves the Human Sciences Badly" - where they lay out their philosophy - they say:

Culture is a part of human biology, as much a part as bipedal locomotion or thick enamel on our molars.

I think this is a pretty crazy position - relatively speaking.