Roundup will kill this batch of weeds. But it won't prevent more from growing in the next few months. Why don't you want to use a weed barrier? It's inexpensive, only labor-intensive for a day (you have to pull back all the mulch, lay it down, then toss the mulch back over the barrier and add a fresh layer), and will prevent a good percentage (not all) of weed growth. After that, you just pull the individual weeds out as they show up.
You'll still have to kill off or physically remove the current batch of weeds before laying down the mat but you'd have to do that whether you have a mat or not anyway, so that part isn't any extra work.
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