We have the "Lifetime subscription" Plan (we don't need to pay for the renewals, this was one time purchase with lifetime updates forever) for the Toolset plugin, and our client's website has this plugin installed.
Their site is currently registered in our 'Lifetime' Plan and using our site key.
The client would now like to manage the website themselves, would they need to setup their own Toolset account and purchase their own subscription to avoid the 'dependence' on our license/site key?
I've inquired about this for you but have yet to get a response. However based on the interface on your profile page it would appear that this isn't possible for Toolset clients.
On WPML there is a button that can initiate the transfer as you say but with Toolset this isn't possible to do.
At present I would recommend that your client purchase their own Toolset subscription in order to ensure that they're getting the latest updates when they become available.