I have been reading a lot online about estimating standardised effect sizes in mixed effects models and it seems like there are formidable challenges even for something relatively broad like an $R^2$, let alone for contrasts. What is a simple explanation (that I could supply to a reviewer) for why effects sizes are so challenging to perform in these more complex models?
There is nothing inherently challenging. What is challenging is getting a consensus on how many degrees of freedom a LMEM has and what an $R^2$ is supposed to reflect in a LMEM. As soon as two parties agree on these matters, the coefficients of determination and effect sizes are "trivial" to calculate. That's why we have all those metrics like $\Omega^2$ (Xu, 2003), pseudo-$R^2$ (Hössjer, 2008, or Nakagawa & Schielzeth, 2013) and approximations like Satterthwaite's (Satterthwaite, 1946), Kenward-Roger's (Kenward & Roger 1997), etc. The Journal of Statistical Software article on
lmerTest offers an excellent discussion on the matter. To that extent, Douglas Bates, the linear mixed effects models' OG, has expressed some of his thoughts on the matter of R2 measure in mixed models online. I fully agree.