Does extraposition work in "I made happy my friend who..."?

I believe this is an example of shifting and not extraposition. Extraposition would require moving a constituent in such a way that it causes a discontinuity, but here there is no discontinuity. If you wanted to use extraposition, you could word it this way.

"I made my friend happy who came here last week."

Please note that the first way you have it written is the most canonical word ordering.

See the Wikipedia articles on extraposition and shifting, linked below, for more details and examples.

• https://en.wikipedia.org/wiki/Extraposition
• https://en.wikipedia.org/wiki/Shifting_(linguistics)