After reading more recent caselaw I think, in practical terms, the reason they attempted a Takings claim is because they wanted monetary damages, whereas because of the peculiar contours of sovereign immunity jurisprudence the remedy for an infringement claim would have at best been an injunction. And apparently more recent sovereign immunity caselaw has cast doubt in some circles on the ability to even get an injunction. I'd personally be surprised it that notion was ratified by SCOTUS, but clearly I haven't been keeping up so my opinion isn't worth much.

EDIT: Just this month SCOTUS agreed to take up the specific issue of sovereign immunity vs copyright: https://www.scotusblog.com/case-files/cases/allen-v-cooper/