Self answering after waiting for two months, From Tourney of Harrenhal, It appears that a Champion has to defeat his co-champions in order to win the right to crown the Queen of Love and beauty.
Total Number of Champions
From Tourney at Ashford Meadow and Tourney at Harrenhal we can establish that there are five champions who defend the honor of the QoLaB.
At Tourney of Ashford Meadow following five were the original Champions:
- Ser Androw Ashford
- Ser Robert Ashford
- Lord Leo "Longthorn" Tyrell
- Ser Humfrey Hardyng
- Prince Valarr Targaryen
At Tourney of Harrenhal, following five were the original champions:
- Four unnamed sons of Lord Walter Whent
- Ser Oswell Whent, A knight of the Kingsguard and brother to Lord of Harrenhal, Walter Whent.
So it is established that total number of champions is five.
The right of crowning the Queen of Love and Beauty
We will use Tourney at Harrenhal to prove the hypothesis here.
Of the existing five Champions, Four were sons of Lord Walter Whent and one was his brother and a Knight of Kingsguard.
Over the course of jousting, Four sons of Lord Walter were defeated by unnamed challengers. Which means, those four victors now took place of Lord Walter's sons as Champions.
According to World of Ice and fire Chapter, "The Year of False Spring", it is stated that Rhaegar defeated four kingsguard. Three of the KG who are known to be participants in the tourney are:
- Ser Arthur Dayne
- Ser Barristan Selmy
- Ser Oswell Whent
The fourth one may have been either one of Prince Lewyn Martell, Ser Jonothor Darry and Lord Commander, Ser Georld Hightower. Jaime couldn't have been there because he was ordered to go back to Red Keep.
From TWOIAF Chapter "The Year of False Spring":
Prince Rhaegar emerged as the ultimate victor at the end of the competition. The crown prince, who did not normally compete in tourneys, surprised all by donning his armor and defeating every foe he faced, including four knights of the Kingsguard. In the final tilt, he unhorsed Ser Barristan Selmy, generally regarded as the finest lance in all the Seven Kingdoms, to win the champion’s laurels.
There are five Champions in total.
Rhaegar defeated four KG and many others. It is logical that some of those may have been Champions at the end. Barristan Selmy and Rhaegar were both champions simultaneously, that much is for sure.
In any case, the transcript refers to final tilt between Barristan Selmy and Rhaegar Targaryen. If this was the final tilt, it means the champions at the end had tilts among themselves, from which Rhaegar and Barristan emerged as the best. In the final title Barristan Selmy lost and Rhaegar Won the Champion's laurels and right to crown the QoLaB.
It is however not clear when do Champions start jousting among themselves. Most logical assumption here would be that they start doing that once all the Challengers have had their day in the field and no one else is left to challenge the final five champions.
NOTE: I will wait for 3 days before marking it as correct answer, so that if any community member has a better answer, he could post it and win the acceptance points for himself.