Does Slicer support this kind of PET-CT registration?

Generally speaking, yes. CT-CT should work very well; PET-CT can be a bit harder because PET is typically low resolution and has poorly defined skull. Please see the Registration Case Library, especially 8, 14, and 20.

I think most of the examples in the case library use other tools, but Elastix is now available as an extension and may be a good option to try first:

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