$request is not interpolated, the whole string is considered to be a method name.

You could wrap it into doublequotes "..." to create meaningful code-text

 eval( "\$client->$request" )

(Untested)

but I see your approach a bit critical, why not simply

$client->DELETE($url)

without any eval?

And if you need to keep the method flexible try

$client->$method($url)

(Untested again)