Before deciding on whether it's okay to move the hedge, you need to know what tree you're dealing with. Assuming that the hedge is, as you note, either hornbeam or beech, then you need to look at the root systems of both trees:
The European Hornbeam (Carpinus betulus) has a wide and shallow root system. It is a fairly slow-growing tree. According to the US University of Connecticut, it's considered difficult to transplant as a large tree. I don't know that I'd consider your hedge to be "large", but this site allows you to estimate a tree's age by entering its diameter; using the American Hornbeam option indicates that your trees, if hornbeams, are about 8-10 years old.
The European Beech (Fagus sylvatica) is a relative of the Hornbeam and like it, has a shallow root system, with many large roots spreading a large distance from the trunk, and is slow-growing. According to the calculator linked in the previous bullet point, your trees, if beech are about 9-11 years old.
Typically, tree roots extend to 150% times the side of the canopy; your trees being pruned as hedges makes this a more difficult calculation. If your hedge is 150mm in diameter (pruned), however, you can estimate that the roots of either species extend to at least 2-3 meters from the trunk. You can confirm this by examining the base of one of the trees - find a relatively large root flaring from the trunk and then carefully remove the soil from above it as it travels away from the trunk. Continue until you lose sight of it, which may be several meters away.
As for the root-pruning/straw method mentioned in the linked post, that can work for some species, but an eight foot fig is far younger than your hedge (fig trees are fast growers). This allows the fig to more quickly replace roots lost to pruning than the hornbeam or beech. You could try it with your hedge, especially if you root-pruned only one side at a time to allow the un-root-pruned site to maintain the tree, but I would wait two years after root-pruning the first side to then root-prune the second side, and then wait two more years to transplant it. Four years is a long time to wait to move the trees.
In my opinion, if you were to move these trees with, say, a 100-150mm root ball, you would most likely kill them. A larger root ball may not be feasible, even with a small excavator. You would have much better success moving the rose with the excavator.
