I'll assume that there is no note before the first and no note after the last, because fingerings will vary based on the preceding note of the section and the note after the last.
You are curious about how fingerings are generated. Let us remember that all fingerings are made-up by someone. In books, they are still obviously made up, but the experienced editor just came up with generic fingerings that will work for anyone in the general public (playing that certain song). These fingerings serve for many purposes. Some fingerings help connect slurs or phrases without lifting the hands (except for repeating notes). Some fingerings help transition your fingers smoothly from one note to another. Some fingerings help reduce the number of transitions (like 3-1 or 4-1) for fast sections. Fingerings also help "guide" your fingers in a passage (like a fast one) to end on a certain note (at the end of the passage or phrase), so it can start the next one (passage or phrase) without having to adjust the fingers as much. Fingerings also help on passages of repeated notes.
These are just some examples.
Please understand that nobody can force you to do a certain fingering. Editors (of songs) are experienced, and choose the fingerings for the easiest and smoothest execution of the piece. When you become experienced, you can also make your own fingerings, but also understand that your fingerings should fit your needs. For example, someone with big hands might use different fingerings than someone with small hands.
Take the chromatic scale (starting on C) as an example. The general way (especially for students) is to go the 131312131313 way, but some people prefer using the 4th finger. I forgot the exact fingering, but it's something like 123412312341. Using only 3 fingers is easier, but using the 4th finger is faster. It depends on the person.
Another question of yours is about avoiding the 3-1 transition. You can correct me, but I'm assuming that your fingers are big enough to make that transition (like you specified). It might feel weird, but avoiding it will give you some pretty big limitations when coming up with the fingerings. The best way is to practice the 3-1 transition until it's not as weird to you anymore.
If you really want to avoid it, do the 123-45- 123-45- 12323- 5 fingering (with the pedal to connect the big 5-1 jump), like nonpop suggested.
NOTE: I will use "-" for note skips, and "()" for transitions (2-1, 3-1, 4-1, or 5-1).
Like Tim said, using your fingering is unconventional (but as mentioned before, you decide).
I would use 12(3-1)2- 341-2(3- 1)2323- 5. I did the 3-1 transition in the beginning to avoid 3-4, which is not much easier than 3-1. Then there is a 2-3 skip, which might be weird, but not so bad. Then there's a not-so-bad 1-2 skip and another 3-1.
Another way is 123-(41) 23(4-1)2- (31)212- 3. The 3-4 skip might feel weird, though.
If you want to avoid 2-3 and 3-4 jumps, you can use 12(3-1)(3- 1)2(3-1)(3- 1)2323-5, but there will be a lot of 3-1 jumps.
Another way to avoid 2-3 and 3-4 jumps is to use 2(31-)2(3- 1)2(3-1)(3- 1)2323- 5.
For my second way, to avoid the 2-3 jump use 123-(41) 23(4-1)(3- 1)2123- 5.
There are lots of ways to come up with fingerings, but- *last time saying this* -the fingering really depends on your needs, so if none of my suggestions work, please specify your conditions (your hands, how fast you play, etc.) and I (or others) can come up with better suggestions.
If you're talking 120 beats per minute, then you may use the suggestions above. If you're talking 120 beats per second (you wrote bps in your question), then you may still use the suggestions above ;)
