I say: ああ、大丈夫{だいじょうぶ}。気{き}にしないで、構{かま}わないです。よろしくお願{ねが}いします。
This response looks okay except for the last part. I am quite perplexed as to why you would want to say 「よろしくお願いします」 in this situation.
「よろしくお願いします」, as a parting greeting, is generally used in situations where you have requested someone to do something. In other words, it often has the nuance of "Thanks in advance!", which is why your other use of the phrase at the very end of your question is appropriate.
(I see users here say ありがとう at the end of their questions, but that makes little to no sense. That is said only AFTER the service has been rendered, or in this case, after an answer has been posted. It is not said in advance like "Thanks!" is in English.)
If, however, you have specifically asked the biker to drive more carefully and s/he has said that she would, you could then correctly reply 「よろしくお願いします」.
Examples of how native speakers might phrase this:
・「大丈夫です(よ)。でも気を付{つ}けてね。危{あぶ}ないから。」
・「ビックリした~。ぎりぎりセーフ!よく見{み}て走{はし}ってね。」
・「ケガはないから、大丈夫。気にしなくていいよ。」
I myself would not use 「構わないです」 as I might sound like I was saying you could bump into me as many times as you wanted.
前向きや!
because their face was probably buried in their phone even while riding their bike.