You were always allowed to place a link at the end of a video and things of that nature. I'm not quite sure what your talking about by putting it next to the youtube name. You mean as a title? That you cant do unless we throughly check whether or not your video qualifies to be "sponsored" by us.
The main idea is to show that while you do support BVEStation, BVEStation doesn't endorse, make, sponsor or manage whatever happens in the video. So as long it doesn't make people think that, you can use the BVEStation name.
Now if you do want that, again we'd have to check whatever your submitting. Usually to qualify for that you'll need to have original recordings, images, and music. The content can also be CC (Creative Commons).
So just take a look on how you want to present yourself and us. I'm guessing you want to suggest that your a fan of BVEStation, then you should clearly show that, most of you would fall into this category, and feel free to use the name with some common sense that shows that. If you want to become a video producer for BVEStation, then you should contact me privately about that via PM.
To answer your question, yes feel free to put the website at the end of the video. You can put BVEStation next to your youtube name as long it doesn't look like its part of an official BVEStation production.