BB sets the draw at a power of 2 less than the total number of teams in a country. The Czech Republic had 3 divisions and 336 teams, so the first round of the cup has 256 teams. I'm not sure what criteria is used to determine which teams are chosen, but but it may be last season's record - based either on finish which would exclude some teams that finish 6th, 7th, or 8th; or wins, which would probably make the cutoff around 10 wins in D.III.
What BB should do is have a preliminary round which would reduce the field to 256 teams. This could be played on the first Thursday of the season. So in the Czech Republic you would have 160 teams playing 80 games, with the winners joining the other 176 teams in the first round proper. They could use the previous season record and bot-ness to determine which teams have to play the preliminary round, so this would include all the bots, and lower finishing D.III teams. The game matchups could still be random, and upper division teams would still start on the 2nd Thursday like they always have.
Under the current system, about 1/4 of all teams are kept out of the cup. In most countries these teams are bots, but in the Czech Republic which was almost full, there are a lot of active teams being excluded.
Since a 4th division was added to the Czech Republic this season, the cup draw really should be expanded to 1024. This would mean all of the existing teams could participate along with several hundred newly minted bots. Since the cup is still more than one week away, this solution may be feasible for the Czech Republic and any other countries that have added divisions.