This is easiest to do in a player match where you can invite your boosting buddy/2nd xbox account. In player match, you can shorten the tournament to 5 rounds as opposed to ranked matches which would be 10 rounds each. Also, in a player match you can control the game type. Set the tournament to Arena style, Flag game type. The flag game type is very short, you can complete governors island in about 2 minutes per round. Dockyards takes about 3 minutes per round and Dumbo can be 3-5 minutes per round, depending on where the flags are placed.
So overall if you are going for the 10 Cash Tournaments + 10 Pickup tournaments + 25 ranked matches, it is much faster to do both tournaments as player matches because you will only need to do a total of 105 rounds, and they can all be flag type. Contrast that with doing the tournaments on ranked, where you will have to do 155 rounds, and you would have to do other game types such as racing or hunter which are much longer than flag.
Note that you do not need to win the tournaments, just complete them.