Rank with Substitution

I have setup a game score/points tracking spreadsheet using the RANK function and it works well for the base points mechanism, i.e.

1st place = 10 pts
2nd place = 9 pts
3rd place = 8 pts etc...

However, the players involved have an opportunity to score bonus "quick start" points if they submit a score for the game within the first three days of starting the game in question's round. I have added a separate "Quick Start" score column to record any scores submitted within the three days.

Now for the tricky bit. The rules require that the quick start score still be within the top three places at the end of the round (14 days) to be eligible for the quick start bonus points. So, each players quick start score (if they submitted one) has to be compared against the final "Base" scores to see if it is within the top three (so a simple IF statement with > should do this). The problem I have is how to individually do this comparison for each player and then decide what rank/position their quick start score would be IGNORING that players final score (this last part is important because the quick start score may be lower than the players final score).

I would like to achieve this without macros.

I have attached my template file and very much hope someone can help.

Thanks