Anyone that gets this and does not submit it is "leaving money on the table." It is an award/recognition that one should be proud of. As mentioned, you don't know a percentile for the Basic exam unless you are top 10% or lower 6-8% (meaning you failed).You will get a letter for the basic if you’re in the top 10% but I have no idea if anyone submits this.
I personally believe that ITE scores do have an influence on how many interviews for fellowship that you will get. When it is time to apply, you will likely only have your CA-I score back, so it tends to have a far bigger weight than maybe it should. Late in the process, you will likely get your CA-II score back in time for it to be considered in your application.
A really high percentile on the ITE will get you way more interviews and a single digit percentile may hamper your efforts. A fail on the Basic is a tough obstacle to overcome, but not impossible. If you are pursuing Critical Care, all will be well no matter what you get on anything. Cardiac, pedi, and pain are significantly more competitive. A program with a good track record of fellowship placement is a big help. Programs that have drawn good candidates from programs tend to continue that pipeline in subsequent years, in my experience. Fellows who go out and are rock stars at their fellowships will really open up the eyes of the fellowship directors to strongly consider future candidates from that program, in hopes that the same skill set will be there. Conversely, I think someone who performs poorly can sabotage residents behind them for the same reason. Fellowships and private practice groups remember people that do an awesome job for them and they remember the ones who suck.