You may be lucky if your client has their name registered as trademark, then contact IG and ask them.. I've read here on BHW that people were able to claim account names that way..
Depending on the name, you may want to try to contact that account. I've done it before, and got them to change their name, so I could change my name to their name (confuse much?

). This was my thought process and approach:
- you don't want to mention that you represent a client (at least initially, if the person on the other side is business savvy, they will smell an opportunity to make you pay for it)
- play as if you are 14 years old ?
Send them a message like this (it made sense for my product, may not make sense for yours but then twist it):
Hello,
my name is [enter name here] and i really like to take pictures of [product name / description].
i would really like to create an instagram page with my pictures of [product name] to show my family.
would you be able to change your name to [productname]1 / [productname]x / [product_name] (you get the idea), so that I can use this username?
that would be so great, i really hope you can help me out
i will follow you ofcourse.
greets
[yourname]
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IF they don't want to change it, you can always approach them with another account and offer them something for the name..