If a uke with vintage style slot bridge has a slight lift on one side, I believe that converting it to a string-thru should prevent it from getting worse, or coming totally off. But, will the conversion close up the very slight gap?
I converted one of my ukes to string through with the thought that there would be less pull on the bridge. I think it's a very logical idea. I used a small plastic washer and beads with a good thick knot. This is Michael Kohan in Los Angeles, Beverly Grove near the Beverly Center
8 tenor cutaway ukes, 4 acoustic bass ukes, 10 solid body bass ukes, 14 mini electric bass guitars (Total: 36)
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I have done this conversion as well, but not for the issue you have. I don’t think the tension of a nylon string would be enough to pull the top up to close the gap at the bridge. You would need to do it to find out for sure.
I personally wouldn't do it. I'd address the real issue, if there is one, of the bridge coming off.
Hard to say without pictures or anything but depending on how much the bridge is coming off or exposed, may be worth while to have a luthier take a quick look. Bridges are usually just held on with glue so is they come off cleanly, it's usually just a matter of gluing back down and ensuring nice even pressure across the bridge while it's drying.
A lifting bridge can easily be repaired with some glue and a clamp. A string through bridge usually requires a bridge plate below the bridge that can anchor the strings. So check with a mirror if a bridge plate is there. If not, installation is likely more complicated than regluing the existing bridge, and may also affect the tone of a soundboard that may not have been designed to have one.
If you use beads it is even more important to make sure there is a strong bridge plate installed. The pressure of beaded strings directly against a softwood (e.g. cedar) soundboard can chew through a thin top quickly. A hardwood or laminate top may resist a couple of years longer.
I’ve not used beads myself but I wouldn’t have thought the flat ended ones would damage much. When I’ve used string through bridges I’ve used very small metal washers under the string knots (big knots for the thin strings) and they distribute the load very effectively.
If you use beads it is even more important to make sure there is a strong bridge plate installed. The pressure of beaded strings directly against a softwood (e.g. cedar) soundboard can chew through a thin top quickly. A hardwood or laminate top may resist a couple of years longer.
Birch is a hardwood that would last longer than cedar or spruce. Probably also depends on actual tension and playing style and the associated level of friction. We can observe damaged tie and slotted bridges that are designed to be under pressure from strings, so an unprotected top is much more vulnerable.
