I would say there are several threads to this issue that you need to consider.
You have a solution ("get my friend into the team to fix the UX") that you are essentially going to try to sell to management, but understand that there is a wider political dimension to this that make create a backlash.
Its always easier to sell ideas if you are focussed on the outcome as a whole, rather than taking credit; this allows you to coach or lead them towards solutions by asking questions as opposed to promoting answers. Of course, this may lead to a different solution than the "obvious" one you are after, since we're not actually engaged in Inception.
The areas you need to think about here are:
- is your root cause of the client issue accepted?
Does the product owner / team leader agree that the issue was a usuability one, or are they focussed on other solutions (client training issue, specific form layout, improving the manual) that may be cheaper or easier to implement?
will improved usability drive revenues?
A lot of companies make money out of product training. A lot of manuals tell you specific details of functionality, without telling you how to perform a workflow, perhaps as a result. Depending on the industry, product and business model improving usability of functionality that is already in place may not be a priority. Fear (losing revenues and/or clients) and greed (selling more product) are generally two of the biggest drivers.
Does your friend want to be involved?
If your friend isn't interested in being part of your team, then its the wrong aprpoach to take. Politics can be a "coverall" excuse for a lot of different issues, including team culture and personality clashes.
How to plant the idea
As a starting point I'd suggest raising the overall issue of usability of the product as a concern with your manager; if you do "root cause" analysis on incoming complaints or tickets this may have been hihglighted.
The best way to do this as a question, or a concern : "Do you think the issues that Joe had over at Widgets Inc. was mainly about usability? It cost us some reputation over at Widgets, and it might have a bigger knock on effect."
If the response is positive, you can move on to "I'm not a UX expert, so while we could fix this based on Joe's feedback I'm worried we have other 'usability bombs' waiting to trip us up. It would be nice to get infront of this - can we get someone in to have a look?"
This is probably as far as you can take things; your friend has been involved in the past, and the line manager will either head in that direction, or they won't.
If your focus is on the product, then it shouldn't matter if it is your friend who solves the issue, or someone else.