I can tell you without reservation that the entire headset, including every race, yes the one on the fork too, has to be from the same set, and the fork race from that set has to fit tightly on the fork tube if you want it to work without posting pictures. Otherwise, you will have to show us what you are working with. When rebuilding my 41 I had endless combinations of all genuine Schwinn parts that will never work together. I learned some things. It is a little easier with postwar headsets, but there are still a bunch of gotchas.
Does locking springers require a larger bearing than modern forks?
If the locking springer is prewar, it takes one of two different completely unobtanium fork races (lower) depending on whether it has a standard or deluxe headset. The bearing size and cages are different between standard and deluxe headsets, and in the deluxe the upper bearing is different from the lower and runs upside down. All are smaller than the postwar lower bearings.
You could probably put an entire postwar headset on a prewar frame and springer, and that would probably work.
If the locking springer is postwar, or more correctly if the locking springer is going in a postwar frame, it takes a special lower bearing race that engages the lock. This special race takes a VSF Schwinn bearing, which has a really funny looking retainer. This is good because the VSF bearing is readily available. It also fits a whole bunch of postwar Chicago Schwinns without locks. The way the VSF bearing's retainer fits in the cup looks upside down. Unlike prewar, the lower fork race that presses on the fork tube is readily available to fit the VSF bearing. There are 2 different size holes the fork race might come with, and I don't remember what they are. You'll need the correct one. The springer takes the smaller one, but always measure. The fork race should fit tightly around the springer's fork tube. There is a raised area at the bottom of the tube to hold it tight. If it fits loose due to wear or damage (pretty common), and it does not get tight on the raised area at the bottom, you will have to make a shim, or you will feel slop. Before pressing it on there, try the fork race and the VSF bearing in the locking cup, and turn it to make sure it is smooth. Look at how the race touches the ball bearings. Nothing should rub on the retainer. If the bearing retainer is in upside down or the parts don't match, you should be able to see and feel it. It is not obvious when you are trying to do it on the bike.
That's the simple version of the lower bearing. Upper is complicated. What parts are you using?