over a long period of time, decay rates will average out even though for a very short period there could be a large change over a short time. The solar flare produces huge amounts of magnetism thought a release process called reconnection.
This is not true: If the theory of nuclear decay is exact, then there will be no variation with time that exceeds the statistic confidence intervall. Hence you have to exactly define what you mean by short in relation to half-time.
In the above mentionned experiments the half-time(s) show different time variation for a "very long time". Climatic influences can be rules out completely. You should read the "paper".