It could run this time next season since all the national cup matches are over by now, the problem with that is that the team that finished first this season would qualify but that doesn't necessarily mean they are the best team in their country anymore.
There are still 4 rounds left in the cup for largest countries. The largest cups have 4096 teams which requires 12 rounds. To fit 12 rounds into 11 weeks of the regular season is one of the reasons for playing two games this week.
An alternate schedule could be to move the national cups to the end of the season, and play the international cup at the start of the season. The two competitions would overlap, with the international representatives receiving a bye into the later rounds of their national cup.
With 74 (and up to 128) countries, 7 rounds are need for an international knockout competition.
In the largest countries, you would have 5 rounds in the NC after the end of the international cup. So you could have 30 teams advancing from early rounds plus the 2 international representatives. The 30 teams would be what is left after 7 rounds, which would let you start with 3840 teams (instead of 4096).
Start of season: 3840 teams + 2 international representative.
After 7 rounds: 3840/ 2^7 (or 30) + 2 international represenatives, for 32.
After 5 more rounds, national cup winner is crowned.
If you added another cup round (there is an open Thursday before the start of the league season), then all 5756 teams in 5 divisions could be accomodated.
In countries with fewer teams, the national cup would start later in the season. For 4 divisions, it would be a delay of 1 week, for 3 divisions a delay of 3 weeks, and 2 divisions a delay of 5 weeks.
Or as others have suggested, the international competition could be played in arcade mode like a PL. This could allow play throughout the season, and might permit group play. For example 32 groups of 6 playing 5 games, with the top 2 in each group advancing to 6 rounds of knockout play could accomodate 192 teams.
Or more radical would be to have the champions of the top league in each country promoting to an international pyramid. There regular season games would then be international. They might continue to participate in their respective national cups.
Or you could have an international cup with all teams entered after they were eliminated from their national cup. Teams that were still in the competition at the end of the season would continue into a second season, playing a 4th game each week.