I also do not know the age of your vise, but I do love Wilton Bullet vises. One day I will get one if I can find one for a decent price.
Your vise looks very nice. A good purchase.
I have a Wilton Columbian style bench vise sold under the Craftsman name. I was frustrated that I could not lock this vise to prevent from swiveling under load. Showing my solution in case this is of future use for your vise.
My original locking screw looks very similar to yours. The front part of the vise removed for this picture.
View attachment 266699
I replaced the original hardware with carriage bolt and all-thread coupler nut. You may need to file the sides of the carriage bolt to get it to fit in your slot.
View attachment 266702
I needed the height of the coupler nut so I could use a normal wrench to tighten the nut. At the time I made this modification I did not have a mill to remove metal from the bottom radius. This is working much better than the original hardware.
View attachment 266703