You are looking at a couple of things causing a volt drop, the natural impedance of the winding and the recharge pulse of the capacitor when a load is placed on the output.
Looking at your measured size of the transformer and the estimated core size, I think a drop of the kind you are seeing is possibly inevitable.
Like I pointed out in the previous post, the larger the cap for a given load, the energy of the recharge pulse (current) is larger, hence you will see a drop on a transformer used around its maximum Va.
BTW, if you have done all this so far, adding a couple of windings is as simple as feeding the wire around the existing winding, if there is room.
Probably 1 layer would do it.
Max.

Does this mean choosing a very large size capacitor (when a smaller size would be sufficient to reduce ripple to the desired amount) is not recommended -- as you mentioned it would result in a larger recharge pulse of the capacitor -- which would be a greater load on the transformer resulting in greater voltage drop on the transformer due to its internal ESR?

 

If you don't have the VA capacity to spare, it pays to design to the minimum % ripple otherwise you can run into the effect you are seeing, or even exceed the VA rating of the transformer.
This occurs when running at what is the maximum design load for the particular circuit.
Max.