You may still get suckers sprouting from the roots beyond where you cover them - it depends on a number of factors. If you do, it's a simple enough matter to snip them off and paint the end with Brush-B-Gon or similar brush killer - which are more for woody plants and more effective than Round-Up. You can also paint the stumps as you cut them with Brush-B-Gon. Rather than covering things with black plastic - which would be effective, but the plastic would never break down, except into smaller and smaller bits of plastic, plus, it would be slippery under the mulch, as well as not letting the break-down of the mulch do anything to improve the soil under it - I would use either flattened cardboard (tape and labels removed) or black-ink newspapers, 4-6 sheets thick and laid out to overlap. (Don't do this on a windy day, and dampen the sheets as you lay them down, so a stray breeze won't lift them. Chasing paper is quite undignified, not to mention time consuming!) They will smother the laurel shoots as well as black plastic, and will break down after a few months or years, depending on how moist the area is kept. You WOULD have to keep all edges covered, as exposed paper will wick moisture, and can last for a year or more - been there, done that.... You would need about 4-6" of mulch. I would imagine that planting in the row would be problematic for a few years, as the laurel roots die and rot, but it could be done, planting between the stumps of the laurels. In which case, I would put down the lesser amount of mulch, and be careful to put no mulch in the couple of inches next to the trunks of the plants. As the mulch breaks down, and to keep it looking good, you should annually add another inch or so to the bed.

Voles or gophers - I doubt a woodchuck would do so.

Looking good Robert.

I have a section of running bamboo in a line with a laurel hedge, the bamboo hasn't run all through the laurel. And if you plant running bamboo there the part where your laurel is isn't the only part it is going to spread into. You will have to install root barrier all around where the laurel is now in order to have the bamboo stay within that same strip later. If you are going to dig all around the laurel hedge and install an effective bamboo barrier - one of adequate depth to be at all reliable - you might as well just dig the laurel out instead. There is a big difference between a new, comparatively thin and short screening planting and nothing at all. You might find that removing the laurel and starting over with something else isn't actually as unsatisfactory as having no screen at all afterward.