Although we may marvel at the intricacy of very small, few things impress us so profoundly as the downright gigantic. Big Fish stories are as old as our species' earliest ventures out to sea, but frustratingly difficult for the biologist to verify. Nowhere is this more evident than when trying to verify the rumoured weight of a gigantic fish. As I'm sure you realise, weight of a given fish can vary enormously over time with such factors as length, sex, health, maturity state and reproductive condition. The more fierce or bizarre a given large fish is deemed, the more wildly exaggerated estimates of its weight become.
Sawfishes are certainly among the most unusual of fishes.
No one knows what such a monster sawfish might weigh. Dubious references to even larger and heavier sawfishes occur. But it is difficult to take stock in such anecdotal records. As you say, few people are willing to haul around a truck scale with them. Is it really that we have fished out the largest individuals, or that verification is so much more readily available nowadays.