When United won the PL, the League Cup, Community Shield and the Club World Cup, while also reaching the CL final and FA semi in 2008/09, they played 20 matches between 22nd November and 31st January.
Should Liverpool progress to the League Cup semi and FA Cup 4th round, they'll play 21 matches between 23rd November and 1st February.
The only difference is the admittedly ridiculous turnaround between the League Cup quarter and their first CWC game, so I'm not really sure why Klopp is moaning about fixture congestion as a problem throughout winter when the only actually daft bit is the Birmingham to Doha turnaround.
Even then, it's not a league game, and they can't be expected to play the same line-up in both, so just play the kids in the league cup and let whoever the u23s coach is take charge, while Klopp and the seniors head to Qatar. Not ideal but hardly like it's going to drastically change the line ups we would have seen if there was more of a gap.