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

In a sense it is the same model, the beta-coefficients are exactly the
same,

Not exactly the same, tough.

KilledPerMillion5Log          -0.62841    0.69872   -0.90   0.3685    
Ruralyes:KilledPerMillion5Log -0.73146    0.07823   -9.35  < 2e-16 ***

Compared to 

                                     NM     bobyqa      nlmin       BFGS
(Intercept)                   4.9857379  3.7283744  4.9477121  3.2138480
QoG                           0.7866227  0.5962816  0.7534208  0.5991817
GDPLog                       -1.5161825 -1.3643422 -1.5111097 -1.2940261
Ruralyes                      4.3436228  4.3422641  4.3419199  4.3415551
KilledPerMillion5Log          0.6632158  0.6005677  0.6276264  0.5216984
Ruralyes:KilledPerMillion5Log 0.7313136  0.7316543  0.7314746  0.7329137    

But pretty damn close to nlmin, which you specifically prefered :-)