You must have a vintage Martin 1T because the newer ones have a tie bar bridge and don't use bridge pins at all.
From your question:
I tried seeing if it actually was small enough to even fit into space between the side of the hole and the side of the pin. It couldn't - it seemed to hang up on the bottom edge of the hole.
That is what it is supposed to do - seat at the bottom edge of the hole. The slot in the bridge pin is there only so the string (not the bead) can fit into the same hole with the bridge pin.
Here is a picture from www.tikiking.com that may help you visualize this.
Note that the bead (labeled string end in picture) is not in the hole with the pin but at the edge of the hole and prevented from entering the hole by the pin.
Since ukulele strings don't come with ball ends you have two options when installing strings on a uke with a pin set bridge.
Option one is to tie a proper knot in the end of the string that is big enough to not slip and to hold the end of the string against the hole through the bridge once the pin is installed. If you use this approach, use a pair of pliers to hold the short end of the string while you pull it as tight as you can so the knot won't slip.
Option Two: Some folks elect to tie a small bead onto the end of the string instead of relying on a knot alone. This makes the ukulele string more like a ball end guitar string.
The potential problem you can run into with the bead (which is less likely to happen with the knot only installation) is that it can get hung on on the end of the pin itself before it seats properly against the inside of the ukulele's top (or bridge plate inside the body) at the edge of the hole. If this happens, it will not be locked in place and the affected string will continually go flat.
You must be sure that you pull the string up so that the bead is tight against the top (soundboard) of the instrument (yours may have a bridge plate under the bridge) - as if you were trying to pull it through the hole. Of course the bridge pin should take up the slack - so the bead won't fit through the hole but it needs to be right up against the hole as far as it can go.
Try pulling the string tight before fully inserting the bridge pin. Once you are sure the bead is right up against the inside of the top or bridge plate, then push the pin all the way in while maintaining tension on the string to keep the pin from pushing the bead away from the edge of the hole. Be sure the string is inside the slot in the bridge pin.
In other words, use the tapered tip end of the bridge pin like a pointing tool to keep the bead from coming through the hole as you pull it tight up to the bottom of the hole. Then push the bridge pin into the hole while keeping the bead tight against the bridge plate on the underside of the top. If you put the bridge pin in tight before pulling the string tight in the hole, you run the risk of the bead hanging on the pin before it is seated properly against the edge of the hole.
After installing the strings - you can use a dentists mirror and a small gooseneck flashlight to look at the strings from inside the body of the uke to be sure all the beads are tight up against the top.
The purpose of the bridge pin is to keep the end of the string (whether knotted or tied to a bead) locked tight against the top (or bridge plate) itself right up against the hole. It is held in place by friction. Sometimes the hole for the bridge pin can get too large and the pin will be loose and unable to hold the string as tight as it should. The easiest fix is to get a slightly larger bridge pin.
If one bridge pin seems to be looser than the others, try swapping them around. Some bridge pins may be slightly larger than others. If that fails, you might try a thin piece of paper in the bridge pin hole to take up the slack. It does not necessarily have to be super tight. The bridge pin is only there to hold the knot or bead against the top and keep it seated and unable to get into the hole.
I think your solution is to try making sure that the bead is tight up against the bottom edge of the hole and not being held slightly away from the top by the pin or not climbing up into the hole with the pin.
If that does not cure your issue - you could also try removing the bead on the C string and using just a knot. Don't forget to stretch your strings gently each time you uninstall and reinstall to tighten all the slack out on both the tuning peg and at the bridge.
Here is a link to a YouTube vid of a Professional Ukulele Technician demonstrating the proper installation of strings on a pin set uke: How to Restring a Pinset Uke And the first ten minutes of this video covers installing strings on a pin bridge uke more slowly and offers more detail on the proper knot to tie: Installing strings on pin bridge uke with detail on knot