I must say that I have two Staggs (got both together for the price of one with some slight damage on both), one has a good fingerboard where I can set the action lower (if it rattles it is on almost any note) and one with a less good fingerboard where I have to put the action much higher. I think the second one has a non-plan fingerboard, so some notes rattle more with a lower action than others.
Since the normal Stagg's fingerboard is painted, it could be a bump in the wood but also a differing thickness of the paint. Sometimes, specially when the action is rather low, a very small bump may cause a buzz. It doesn't need to be visable. In this case the board/paint below the fingered note where the buzz started nedd to be scraped off a very little bit. But since you can make things worse, if you don't know how to start and proceed, this is not something to be done by an unexperienced DB DIYer.
If you want to try it, start at the lowest fingered note that buzzes, scape until the buzz vanishes or starts as early as normal on the fingerboard, and then proceed to the higher fingered notes. And don't stick to the notes of the chromatic scale, this is a continuous pitch thing, halftones are too much spaced to get it right. But without scooping the fingerboard, you only can make it a little bit better, it won't solve the real problem.
Other strings might help too. I got a used set of Presto Balance Hybrid Medium (polish strings, that are available in Germany too, but not in the USA) and they fit well on the bass, have a little bit more tension, but the rattling is just a little bit less. But nicer strings than the stock ones. They are also available in light tension.
I'm afraid the Stagg has to be played lightly, not by digging into the string too much like I would do on my acoustic bass or my Clevinger EUB, which both have very nice fingerboards (after some planing).
Since the normal Stagg's fingerboard is painted, it could be a bump in the wood but also a differing thickness of the paint. Sometimes, specially when the action is rather low, a very small bump may cause a buzz. It doesn't need to be visable. In this case the board/paint below the fingered note where the buzz started nedd to be scraped off a very little bit. But since you can make things worse, if you don't know how to start and proceed, this is not something to be done by an unexperienced DB DIYer.
If you want to try it, start at the lowest fingered note that buzzes, scape until the buzz vanishes or starts as early as normal on the fingerboard, and then proceed to the higher fingered notes. And don't stick to the notes of the chromatic scale, this is a continuous pitch thing, halftones are too much spaced to get it right. But without scooping the fingerboard, you only can make it a little bit better, it won't solve the real problem.
Other strings might help too. I got a used set of Presto Balance Hybrid Medium (polish strings, that are available in Germany too, but not in the USA) and they fit well on the bass, have a little bit more tension, but the rattling is just a little bit less. But nicer strings than the stock ones. They are also available in light tension.
I'm afraid the Stagg has to be played lightly, not by digging into the string too much like I would do on my acoustic bass or my Clevinger EUB, which both have very nice fingerboards (after some planing).
I really don't think it's going to hurt the fork, or the bass... what I suggest is getting a small, thin bungy cord, and wrap it around the end-pin nut, and the stand to keep it secure, or a piece of velcro instead. Either one of these should work for you, and tell your children they can't step in Daddy's four-foot Stagg bass safety zone! 
