I've got 4 badges on my badge bar, I've never noticed any detrimental impact on cooling. I think it might have been an issue on the old Plus 8s, their cooling was somewhat marginal at times, but for a recent Plus 4 I can't see it would be an issue.
Tim H. 1986 4/4 VVTi Sport, 2002 LR Defender, 2022 Mini Cooper SE