Far more effective then any chemical is physically removing the grass, roots and all, from those beds with the added benefit that you do not pollute your environment.
There are poisons that you could spray that might kill the grass, leaving ugly brown grass blades that will need to be removed anyway, but none I would use or recommend.

Some of the herbicides are not very effective getting rid of the roots so after spraying the top growth and killing that the grass grows back requiring respraying. If the roots are dug out, all the way back to the edge of the bed regrowth will take much longer and you will not need to spend as much money on a "solution" that is not all that good.

Ortho Grass-b-gone kills grass and not some ornamentals but I read the label and it says it might hurt some junipers. It says to spray around and under the plant but not on it. Grass growing in groundcovers is very troublesome-it doesn't always pull easily and just regrows.

Grasses, and many other weeds, will regrow if you do not get the root system out of the area. Many of these plant killers wipe out the top growth but not the roots so you need to keep spraying them while a good digging out of roots will keep them from returning for much longer.

Just wanted to warn people about getting a rash from working with/around juniper. It gives me a poison-ivy-like rash that is much more short-lived but still very irritating and gross-looking for a few days.

Knowing how the grass is getting into the juniper would be good to formulate your defense strategy. If it is creeping in from the yard, you will need a border barrier to make it stop doing that. If it is coming from seeds, you may want to mow more often and/or mow with the clippings going away from the juniper.

If you are able to add new mulch, as you mentioned, you could use newspaper or cardboard as a smotherer, which would be easier to work with than a weed fabric or plastic. And newspaper or cardboard would not end-up creating a bigger problem if/when grass starts growing again. There is no need or reason to remove the existing mulch first. If you put an inorganic barrier under the juniper, grass and other weeds can still germinate in the mulch on top of it. Then you're back to square one, only then you would also have the fabric or plastic to deal with. Eventually, the soil under an inorganic barrier will become devoid of nutrients causing the juniper to be unhealthy.