When looking for a copula I could simulate from (rather than the Gaussian copula), I found an algorithm for the Ali–Mikhail–Haq copula
that was proposed by Kumar (2010) as reproduced above. But the method seriously fails in that the range of (U,V) resulting from the simulation does not even cover the (0,1)² square! Unless I made an R coding mistake (which is always a possibility).
sim=function(T=1e3,h=.5){
o=matrix(runif(2*T),T,2)
a=1-o[,1];b=1-h*(1+2*a*o[,2])+2*h^2*a^2*o[,2]
d=1+h*(2-4*a+4*a*o[,2])+h^2*(1-4*a*o[,2] +4*a^2*o[,2])
o[,2]=1-2*o[,2]*(a*h-1)^2/(b+sqrt(d))
return(o)}
There is no explanation in the paper as to why this algorithm is (not!) working, besides the inverse cdf argument—with the reference in R copBasic temporarily worrying until I checked the cdf inversion is completely numerical—, but a correct version can be derived from inverting the conditional cdf of one component, V, given the other, U. Namely, since the conditional cdf is given by
which leads to a second degree polynomial equation (in v) when solving the equation F(v|U=u) = w.
sim=function(T=1e3,h=.5){
o=matrix(runif(2*T),T,2)
v1=o[,1];w=o[,2]
d=(2*h*v1*w-1-h)^2-4*(1-w)*h*(1-h*w*v1^2)
o[,2]=-2*h*v1*w+1+h-sqrt(d))/(2*h*(1-h*w*v1^2)
return(1-o)}
And with a more likely outcome (Xed checked by comparing F(u,v) with its empirical version for several pairs (u,v)):
Managed to 
Nicola Branchini presented his work with Victor Elvira (a close friend and coauthor, incidentally one of the organisers of the workshop!) on improving self normalised importance sampling by interpreting it as a ratio of estimators based on two samples (which may be the same) and attempting to optimise the joint distribution of said sample. The starting assumption is having (good) marginal importance functions, which means the goal here is in optimising a copula distribution targeting the ratio as quantity of interest. Optimality is however defined in terms of the approximate asymptotic variance of the ratio, which remains an approximation. The idea is nonetheless quite interesting and shows potential for connecting with bridge sampling and… 
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The 
