I'm wondering if there's a way to use bridge weight as a restriction. Would I need to just load points with that info, or could it be included as an added field in the road table?
Go for parametirized attributes (add a field with the weight for every road link that is used for the network dataset; in this case you might dynamically choose what is the weight of the vehicle that is used currently for the routing)
Personally, I think you have two choices here:
Go for parametirized attributes (add a field with the weight for every road link that is used for the network dataset; in this case you might dynamically choose what is the weight of the vehicle that is used currently for the routing)
or
barriers (convert the road links that represent bridges to points/lines and use them as scaled cost or restriction barriers; problem here would be that if the truck weight can be of various values then you have to specify what restrictions will be applicable for each routing case).