It took me a minute to understand the problem, but I beleive you are saying the falsework shown on the bid documents was to support the girders and not the wet concrete for the deck slab.
I assume the bridge is closed to traffic for this work and we can ignore live load considerations.
If my assumptions above are correct, then the designer is simply asking for these existing cast-in-place girders to be shored during the replacement of the deck slab. If the girders are shored at midspan, much less stress will be put into them from the wet concrete for the new deck slab.
After the concrete deck has cured, the shoring can be removed and much more composite capacity will remain for the live load than if the shoring did not occur. Even if the original girders were not shored during construction of the original deck, by shoring for the replacement, the live load capacity can be improved somewhat.
The shoring also prevents excessive deflections at midspan which could also have been a design consideration.
I hope your bid price included enough for the shoring, and the shoring should be designed to support the weight of the wet concrete, reinforcing, forms, screeding equipment etc...
The shored girders act as a two-span continuous beam so make sure the reaction used for the design of the shoring towers takes this effect into account.
Good luck, and give us an update on the falsework used once it is designed.