Will the herbicide do the job. I think that this is the main question. If it works then either use it or not. I live with this problem as well but have avoided using any herbicides, except on plants in the cracks of sidewalks.
My approach is work the ground up and remove all that can be removed and then next cycle work the ground up and remove again. I have been able to force it into inaccessible spots or under large shrubs that I cannot dig around.

If it works and it kills some of your other plants you can always replant later when the herbicide has been dissipated.
Just be sure to use it before the grass seeds unless it has a residual effect in which case you will either have to wait awhile for replanting any damaged plants or buy plants that will not be affected.

It is a tough choice but which is preferred??