Every region has types like that, native or introduced. Plant one and it won't stay just one.
I have native hackberries (oak family) growing around my fence line, and if I left them alone, I'd have a forest of them. Anything other than mowing, hitting with the string trimmer, and lopping off bigger ones would be expensive and probably toxic to that area of the yard. If that isn't going to do it, then your barrier sounds like the best approach.
This is just an educated guess: a barrier like you suggest has been used down here by cities to keep roots from pushing up sidewalks or streets when planting street trees. A dozen years ago when I helped promote a street tree program in my neighborhood the City of Fort Worth was doing that with panels of 1/4" to 1/2" plywood, it didn't need to be a metal barrier. They were placed adjacent to the tree, the depth about 2-3 feet to force the roots down and away from the zone directly under the paved surface. They were plywood because after they broke down it wouldn't matter, the tree in question would have deep roots and not need a further barrier.
So how deep are the poplar roots? How far down would a barrier have to go before the roots that grow beneath it stay low, not becoming suckers or rhizome type surface roots? How established are the suckers in your yard already? If you put in a barrier you'd still need to keep the existing roots down until they die off.
My back aches just thinking of the work it is going to take to dig that kind of trench, especially if it is full of tree roots already! This is kind of the opposite of the Robert Frost poem "The Mending Wall." The trees in your neigbhor's yard will attack the trees in yours.
A good fence (with a deep concrete footer) would make for better neighbors. Maybe you need a deep concrete-filled trench along the property line.