I used the this image, the anime versions are longer and inconsistent.
A slightly thinner snake at .75m weighs .1kg. Compared to that snake, Shenron's ~350m length would be 11,200+ tons.
The question is whether or not you can use the volume of a snake to scale it. Because Shenron's head is proportionally smaller than a snakes.
Besides, I'd like a little more to go on than a mere guess for the length. You could use some character and copy paste him or her along the center of Shenron's body, count them, and find out the length that way. But it would be a long and tedious process.
You could also manually put down points on Shenron's body in MATLAB, spline them, and curve-integrate that numerically for optimal precision, but that would take even longer.
Its head is roughly 6.5x the height of the character below, who I believe is Yamcha, of 1.83m and is <1/30th its total length. ~12m head, ~350m body.
The snakes body length is measured at .75m and head at 0.025m. It has give or take the same proportions as Shenron.
Its a rough calc, yes. But there's no need to waste time getting a specific to-the-mm result as only a general result is needed to answer the threads question.
But now you're using the potential error of the snake's bodily relation to Shenron's, twice, and that's not helping the fact that the first one is of the third order.
Either way 10,000 tons doesn't seem to be too far off. Either way, if you take Goku's 40 ton-shadowboxing into account he'll probably won't be able to lift Shenron.