An effect similar to sidewall thickness can be seen in BRR’s inner tube testing. The effect of a Conti Cross 28 butyl inner tube, which has an uninflated rubber thickness of about 0.6mm according to my calliper measurements, produces losses of 2-6W per wheel versus tubeless, depending on pressure and tyre width. The tyre width will affect how much the inner tube stretches, so the tube thickness when inflated, hence the variation in losses.
Those losses are coming from the extra hysteresis losses of the additional rubber, similar to what happens when the tyre sidewalls are thicker. Interestingly though, that also seems to be ~1W per 0.1mm, that order of magnitude.
For tyre sidewalls, the losses will also depend a lot on the rubber compound and construction of course, because we see big differences in the BRR number for tyres from the same brand and model with different compounds, different TPIs, etc.