How is the "best loser" determined in football?

I would say the third team, since they lost on a penalty shootout, thus implying that they were able to hold on to a draw throughout regulation and extra time. As for the other two teams, I infer from your post that they lost in regulation outright.

You're creating a problem that doesn't need to be solved, when better methods exist.

A quarterfinal round can be complete with only six teams. This naturally requires two byes, and it makes sense to have those be the two highest-ranked teams from pool play.

They are crossed in semifinals such that the first-ranked team plays the lowest-ranked quarterfinal winner, and the second-ranked team plays the highest-ranked quarterfinal winner, as done by a number of professional competitions.

Given that the games have already been played, the fourth place should go to the team with the best performance in the "quarterfinal" which is the team who achieved a draw, not either of the teams who were beaten.

Ideally, the rules of the tournament should be defined before it starts, but now that we're here:

In my opinion, the best loser out of 3 teams lost in the quarter-finals (or 1/6 finals in your case) is the one who performed better in the games involving only those 3 teams during round-robin phase or even who finished higher in the table.

Because this way the loser will be most probably the strongest one. Why? Here's a demonstrative example:

It would have been better if there was predetermined criteria to select best loser. As there is no indication of that criteria, selecting best loser can be done in following ways:

1. Taking FKL Cup - Best loser rule : Since third team lost in penalties as opposed to other who were beaten in regulation time. So third team qualifies as best loser.
2. Looking at points or position after league matches : Among those three teams which performs best in league matches progresses to next round.

In my opinion you have to set quarterfinals with the best 6 teams as following: