Where do the words Bike and Trike come from? If you are going to abbreviate, why not “Bice”?
This is what wiktionary has to say about it:
Because “bice” rhymes with “mice.”
terminal -ce in English has a soft “s” sound, not a hard “k” sound.
I assume the OP knows that, and is asking why bice-that-rhymes-with-mice isn’t the natural abbreviation for “bicycle.”
To the reasonable possible explanations that engineer_comp_geek quoted, I’ll add a WAG that “bice” might have been avoided because it sounds a plural.
Why is it such a mystery? It follows the same form that other word shortenings do such as cuke for cucumber, nuke for nuclear, and even though it gets misspelled more often than not lately, mikefor microphone.
Except that all the abbreviations you mentioned have a hard “c” like “k” in the original word, whereas “bicycle” has a soft “c” like “s”. Why use a “k” in place of an original “s” sound?
Similarly, the hard “g” in (excuse my French) “faggot” or “moggie” is preserved in the abbreviations “fag” and “mog”, while the soft “g” in “refrigerator” or “vagina” stays soft in “fridge” and “vag”.
So why, as the OP asks, would the abbreviation process change a word that sounds like “bice-uh-kul” with a soft “c” to “bike” with a hard “c” sound, instead of “bice” with the original soft “c” preserved?
The same alternation occurs in other words pairs: medicine/medical, criticize/critical, electricity/electrical. (There is also g/j alternation as in allege/allegation.) Although it can’t be a full explanation, speakers may have a high-level model of some consonants with the sound dependent on context (especially the following vowel).
Thank you for the topic - it’s obviously a weird thing, I’ve been seeing and hearing it all my life, and yet I’ve never thought about it.
I started to try to construct a rationalization of the situation based on how the word “cycle” is in Greek, and realized my rationalizing was pointless because the question is how did it really get that way, not how could it have got that way. 
Why don’t we call a motorcycle a moke?
In the tarantula collecting world, there is a genus of tarantula from India called Poecilotheria. It’s pronounced PEE silotheria (although it gets pronounced POE a lot). The nickname for this genus is Pokie. It comes from Greek, as well.
Let’s start doing so today
It ruins the Guthrie song with the pickle, though.
Because Sir Alec Issigonis, the guy who designed the original Mini already used ‘Moke’ to describe an open top, cut-down version of a Mini van that was manufactured at Longbridge in the 60’s and later in Australia and Portugal. I read that there is now an electric version produced in the US of A.
https://pvlimages.blob.core.windows.net/images/verylarge/DPN57C_27032018_58.jpg
‘Moke’ was also, apparently, a diminutive for donkey.
Maybe the “k” sound comes from the second c, not the first one? So it’s bicycle -> bicyke -> bike.
While we’re at it, “bike” can refer to either one, but seems to more often mean a motorcycle, even though “motorcycle” doesn’t have a B anywhere in it, but a “cycle” is always the pedal-powered one, even though both words contain “cycle”.
Maybe it’s just that words with hard vowels at the end sound shorter and more “self-contained”, and the whole purpose of abbreviation is to shorten words.
Gives a whole new meaning to Gumby and Pokey.
The word “cycle” comes from the Greek “kyklos”. So “bicycle” should be “bikikle”, rhyming with “tickle”. And the short form should be “bike”.
IANALinguist, but something like this feels right to me. I think also the ‘k’ sound is hard and is sort of the essential sound in ‘bicycle’. The s and soft-i are so light they can vanish and you still have almost the same word. So it’s bie -si -kel >> bie -kel >> bie -ke
The Greek-origin idea and the logical-shortening idea are both good ideas… but we have no evidence that they really are why.
That’s one Pokie I don’t want to collect, mon.