As far as I can tell, Twitter media filenames are ever-increasing (monotonic) numbers, encoded with letters.
The encoding is called base58, and DosAVAwUUAAVGQ8 is just a more compact way of writing 182269945157265620266146607.
The way to get a specific pattern of letters is to post content at a specific time.
It's relatively easy for the first few letters (with enough patience). DOITQ had its moment.