If the permit routes you on that road I would say you are OK. It would be nice to know if there is a light bridge down that road. Maybe try and contact the local Sheriffs ofc. If you are talking about an annual permit, no I would not go down that road.
Are you talking about an Extended Weight Permit or a one time Over Weight Permit. If is a one time Over Weight Permit, it should have an exact route for you to take. If it is an Extended Weight Permit, which is an ongoing over weight permit, the state will have a map of the roads that you are allowed on.
In Texas, a Weight Tolerance permit issued by TxDOT is not valid on the interstate (and possibly on non-interstate highways). I think that it will allow you to exceed the posted weight limit on a load-zoned state highway, farm road, or county road, assuming the county you're in is listed on the back of the permit.
I don't know if the permit will allow you to exceed the posted weight limit on load-zoned bridges. You'd best check with a License & Weight trooper for a definitive answer.