I believe you will find that this is related to the JIRA I raised recently.
The issue you are having is fairly fundamental to the system and can cause a lot of frustration and confusion. I would expect that this sort of bug be addressed fairly quickly.
Unfortunately it appears that most(all) of the Seam guys are so caught up in Weld or whatever it is getting called these days that very little is being done to address the issues that a lot of us are having with the existing library. I've raised 2 JIRAs over the past month or so and neither of them have received any attention. Not even a comment. My guess is that the only way these issues will be fixed is if you prove that they exist in Seam3 or wherever the time is being spent at the moment.
Anyways, enough of my ranting.... here's how to solve your problem.
For the time being, to work around your issue don't use Injection. Instead access your Seam bean direcly using the Component.getInstance(class) method. This will avoid the variable being disinjected at the end of the invocation of your method.