Ok will try again without mentioning b**a t***l.
I have the old sky go app for tablet (downloaded from 1 mobile market) working on my 3rd Gen HDX 7inch. It won't work on my wife's 2nd Gen HD 7 inch. I can speculate that it knows what the 2nd gen device is and so says it is incompatible but the HDX was released later and so it only identifies it as a generic Android Device and so works.
I cannot comment on whether or not it might work on the newer 4th Gen Kindles so would suggest you give it a try if you can still get hold of the sky go for tablet app. It may work.