I am wondering how you would translate the short i sound in words like bin, fin, thin into katakana. I am trying to translate my name which contains this sound (with a preceding r). I don't think ィwould be accurate here, but I'm not sure what is used conventionally.
Japanese has exactly five vowel sounds: //a//, //i//, //u//, //e//, and //o//.
English has eleven or twelve, maybe more, depending on your dialect, as cataloged here at Wikipedia.
Since 11 > 5, you're going to have some vowel sounds in English that just don't have an exact match in Japanese. The best you can do is find a reasonably close approximation.
For purposes of the "short I" //ɪ// sound like in English thin, the closest match in Japanese is the //i// sound as spelled using the kana starting from イ. This vowel with a preceding "R" would be spelled in katakana as リ.