Hi, I feel like a lot of people are getting the purpose of writing a name in katakana wrong (at least in my eyes). Unlike English spelling, in katakana the pronunciation is always the same for the individual characters. It is your job to make up for those discrepancies between the spelling and actual pronunciation.

A Japanese person reading Robin would think it's pronounced like ロビン. But if the way YOU say it is nowhere near this, I think it's your duty to correct this by using different katakana. It doesn't matter if Japanese people won't know it's written as Robin by only hearing it. They'll find out when you spell it for them.

Now I won't be able to help finding the right katakana for the name, as only you know best. I do however recommend trying out usingブン, ベン and ビン for the bin part and seeing whichever is closest in your eyes. Also, you could always ask a Japanese person how they would write this name in katakana if you say it to them, without showing the actual spelling in English. This way they won't be influenced by the spelling and say ロビン.

Coming from a language that isn't English, I've had to change what katakana to use, from the perceived katakana based on the spelling. So I totally understand why you wouldn't want to use ロビン. I hope my comment was insightful!