The legend of the Chinese dragon
The accepted answer is simply wrong.

It is probably from a Chinese folk tale since it claims to be Chinese. I daresay I have read similar folktales, especially the swallow-the-bead-and-turn-into-a-dragon part, when I was a kid. I cannot figure out the exact one. But it is easy to search out folk stories with similar motifs.

e.g. https://book.douban.com/subject/11633147/


  一个叫阿宝的孩子拾到了一颗红色的宝珠，他把宝珠放进米缸，米缸里立刻盛满了白花花的大米；他把宝珠放进钱罐，钱罐里立刻填满金灿灿的钱币。阿宝用这颗宝珠帮助周围的邻居都过上了好日子。奇怪的是，自从阿宝捡到这颗宝珠后，天上就再也没有下过一滴雨，大地干涸、庄稼干枯，乡亲们愁眉不展。
  
  一天晚上，阿宝在梦中见到一条雨龙在空中喷雨，醒来后，他决定动身去寻找这条雨龙。一路上，阿宝帮助了巨蛇、梅花鹿、鲤鱼和苍鹰，每个动物朋友都赠送了他一样礼物，并警告他要当心一个叫红鬼的妖怪。随后，阿宝果然遇到了红鬼，并从红鬼口中知道了自己手中的珠子叫龙珠。为了保护龙珠，找到雨龙，阿宝果断决绝地吞下了珠子。最后，在动物朋友们赠送的礼物的帮助下，阿宝变成了一条会下雨的龙，把甘露洒向了人间。


沙县民间故事
This story lacks the part of the magical power of the dragon bead, but retains the part of swallowing the bead to become the dragon.

https://www.sohu.com/a/349852173_488223

This one is rather long, but the story is almost the same, which is easy to find if you can read Chinese. 

A further question is whether this kind of folktale has a origin in Classical Chinese literature.

And I must say many Chinese folk tales have a style and sentiment that you won't find in classical Chinese literature. Not to say even China proper is a country that comprises several distinct territories, varied cultures, and peoples. It is always hard to say what is really "un-Chinese".