The problem is solved. (Please change it to BC2 in the title, I made a typo because I meant DXT3).
The solution to the problem, as mentioned above, lies in the incorrect color. However, I will explain it in more detail for future reference.
1) The bottom layer (background) should be black, and the texture is drawn on top of it.
2) Select the entire texture separately from the background and create an alpha channel; fill the selection with white.
3) Save it in BC2 or another format. Disable MIP maps.