lme4, failure to converge with a range of optimisers, trust the fitted model anyway?

You also still need a random effect for the cluster.

I think I've just stumbled into something that may deepen my
understanding of mixed-models, thanks to you.

I took your advice on how to create the dependent variable

cbind(y, n-y)

and included the random term for cluster,

Formula: cbind(Deprived, Not.deprived) ~ (1 | Country) + (1 | ClusterID) + QoG + GDPLog + Rural * KilledPerMillion5Log 
Data: my.small.df

and fitted the model to an aggregated version of the original data
set. However, while so doing I thought: "this will be exactly like the
model that glmer could not fit without warnings, I'll have exactly the
same warnings again".

In a sense it is the same model, the beta-coefficients are exactly the
same, but in another sense it apparently is not, the warnings are gone
:-)

I guess the difference is that glmer does not have to care about
residuals at the individual level anymore.

I now understand this model as a kind of repeated measures model,
where each cluster is measured repeatedly, once for each individual in
the cluster. While that tecnically does not describe how the data was
generated, it is a clever shortcut to get what I need. Thanks again!