The problem is solved. (Please change BC2 in the title, I made a typo, as 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 very bottom layer - the background - should be black, and the texture is drawn on top of it.
2) Select the entire texture, separate from the background, and create an alpha channel. Fill the selection with white.
3) Save in BC2 or another format. Disable MIP maps.
However, the question of whether something can be done about the dark, semi-transparent background remains unresolved. Due to the alpha channel, some elements become slightly transparent because of the black color in the texture. This is not critical for me, but I doubt I will be able to figure it out myself.
