Actually no - letting her have the toy will help with possessiveness. Carrying the "prey" is all part of the prey drive, and frustrating the prey drive (like not allowing her to possess her catch) only increases the drive to possess the prey.
We do this with young dogs in bite training - they are allowed to carry the bite-pillow or sleeve as a reward. Constantly making them give it up can create conflict which will affect the "out" command at a later stage. Of course, when you give an out command, the dog should obey - but set up your dog for success by not asking for too much too soon, and setting yourself up as a provider of good things (providing the ball or tug to play and carry), and not setting yourself up as the "taker-away" of good things.
With time, as your dog becomes more secure in knowhing that the toy will not be taken away and cause frustration, she will more easily and willingly give up her toy for you.
Some working dogs also need to be working all the time, and to some carrying something is work. I often go for longer walks and let Gryffon choose a toy and bring it, and he'll carry it all the way. I think a normal walk is just too boring for him, having a job to do, like carrying something, helps occupy his brain and keeps him focused on a task.
Bonus is we have a ball or tug for interaction, so I'm having fun too.