If the track turns but a cart moves at a high (derail) speed it will continue straight ahead unless there is turning support (such as a wall or a bridge) in the way. However, a drawbridge has a 100 tick delay in reacting, so you'd have to have the pressure plate placed a fair bit before the cart reaches it. I think a closed door works as a turning support, and it reacts immediately.
There are a number of people on the board who have an excellent understanding of how mine carts work in detail, but I'm not one of them, so you may want to wait for better advice.
However, setting this up is likely to be rather tricky, so I wouldn't do it unless getting it to work was an important focus of the fortress.
Sorting on weight obviously requires fine control over the weight, which means control over cart material as well as all transported contents (think booze in barrels and pots made of a number of different types stone and wood). In addition to that, it would probably work only if the level to which the mine carts are filled is tightly controlled.
If I was to try something like that (which I'm too lazy/not sufficiently interested to do) I'd probably try a setup where the mine cart's starting position determined the network configuration (a bit like an old fashioned telephone network connection when a call was made). Basically, a pressure plate passed over by the cart would set up the control bridges in the configuration for that starting point, while blocking all other entrances (this would require memory circuits to ensure each drawbridge in the "active" position remains "active" even when the pressure plate off signal is sent). Once the cart reaches the destination the other entrances are opened (this needs to be designed such that you don't get multiple carts entering as a result of opening the entrances). I'd expect a lot of debugging as well.