It's not about the weight. MK has and uses more heavy guitars than the National.
According internet, the most heaviest National, weights 9 Lbs. Actually the National Style O weights 8. The Gibson Les Paul guitars weights between 9 and 12 Lbs.
MK plays most of the songs standing and with Les Paul guitars so, no, the weight is not the reason MK is not playing his National Style O anymore. I insist that he said in the BBC Piano room show that it has deteriorated and he doesn't wants to use them live anymore to avoid more harm.
It's not me who says it. It's Mark Knopfler.