Confidence interval for relative contribution of random effect variance

Now, apart from this aspect, can confint be tweaked to calculate not
only the confidence interval of the 'raw' parameters but also for
some function of the parameters? If not, do I need to move to an
implementation using MCMC methods (MCMCglmm, Bugs-type of
approaches, STAN or Laplaces-Demon) to reach my aim or do you have
another (simpler) suggestion?

  You can compute parametric bootstrap confidence intervals of
any quantity you want by applying boot.ci() to the results of bootMer()
(bootMer()'s second argument is the summary function, which you
can define however you like).  This is computationally expensive,
though (even more expensive than MCMC-type computations).

  In principle you might be able to use likelihood profiling
(which is what the default confint() method uses) to compute
profile likelihood confidence intervals of arbitrary quantities,
but you would need to be able to constrain an optimization algorithm
to the specified values (i.e., you would need to set nonlinear
equality constraints; there are functions in nloptr and elsewhere
(many of them called auglag()) that implement an augmented Lagrange
multiplier algorithm for such constraints, but I haven't tried it
out to see how it works.

The advantage of parametric bootstrap/MCMC approaches is that
you also get a finite-size-appropriate result; likelihood profiling
would inherit the asymptotic assumptions of the likelihood ratio test.

glmmADMB still implements a post-hoc MCMC sampling strategy simpler
to mcmcsamp (but you would be on your own for making sure the
chain was well-behaved, etc.)