The teams would enter their national cup after they were eliminatedfrom the international cup.
therefore screwing up the schedule of the national cup.
Not really. Let's say that the international cup had 256 teams, which first played a 4-team group stage (3 games), with the first place team in the group advancing to a 6-round knockout to determine an international champion. So that is a total of 9 weeks. And let's further say that the international competition permitted up to 16 teams per country.
Now consider a national cup with a nominal field of 8192 teams running concurrently. When the international cup is finished, the national cup will have been reduced to 16 teams. But since it is possible for 2 teams from that country to play in the international cup final, only 14 of those teams could have qualified from the national cup. Similarly, the 2 semifinal losers would have entered the national cup the previous week; 4 quarterfinal losers the week before that; and 8 octofinal losers the week before that. So our field for the national cup will be:
Week 13: 2
Week 12: 4
Week 11: 8
Week 10: (16) 14+2
Week 9: (28) 26+2
Week 8: (52) 48+4
Week 7: (96) 88+8
Week 6: 176
Week 5: 352
Week 4: 704
Week 3: 1408
Week 2: 2816
Week 1: 5632 teams.
5 divisions have 5456 teams, so in the first week, not all teams would play. We need to eliminate (5632-2816) teams or 2640 teams. To have 2640 losers, we have to have 2640 games pairing 5280 teams, giving byes to 352 teams.
Now imagine that 8 of the national teams were eliminated from the international cup at the end of the group stage. If they are immediately re-entered in the national cup, there will be too few teams left in the national cup.
We could let them play scrimmages until Week 7. But one reason for excusing them from the national cup was that if they were competitive in the international cup, they would have likely won a national cup match that same week. But since they were bounced from the international cup, we can't assume that they would win in the national cup - plus they shouldn't miss out on game revenues.
So we need to readjust the table above.
Week 13: 2
Week 12: 4
Week 11: 8
Week 10: (16) 14+2
Week 9: (28) 26+2
Week 8: (52) 48+4
Week 7: 96
Week 6: 192
Week 5: 384
Week 4: 768
-----------------------
Week 4: 704 + 8
Week 3: 1408
Week 2: 2816
Week 1: 5632 teams.
Since we have completed 3 rounds of the cup, we only have 704 teams left, plus the 8 teams re-entering from the international cup. But we should have 768 teams under the adjusted schedule.
There are two straightforward solutions: Give teams byes in the 4th round so that the 712 teams are reduced to 384 teams (this would mean 56 byes).
Or alternatively, we could have lucky losers from the 3rd round be re-entered into the cup so that there are 768 teams competing. We had 1408 teams playing 704 games, so 64 would need to be re-entered. They could be chosen by lot, or perhaps based on the closeness of their loss.
Note: In countries with fewer than 5 divisions, the start of the national cup would be delayed so that all would finish at the same time, rather than start at the same time. This would be especially helpful in new countries since it would permit more of the newly established teams to compete in their national cup - instead of being eliminated as bots.
The number of games played in each round, as well as the lucky losers or byes is readily computed.
If the international cup games are not played at a neutral site, they should be played at the ordinary game time for the host. Since national cup games are scheduled soon after the previous round is played, the international cup games need to played at earlier day (perhaps Wednesday or Monday).