Because bacterial wetwood is not fatal, and most trees tolerate it quite well, it is usually more of an aesthetic thing. Coppicing, while it may not kill the tree, will cause a good deal more stress. For your information, siberian elms are very resilient, and will likely survive, but this won't decrease the chances of them getting bacterial wetwood in the future, as it is a sap infection. It won't go away if you cut the tree. See here.
Bacteria, commonly found in soil and water, take up residence in young trees or gain entrance to older trees through wounds. The bacteria, including species of Clostridium, Bacillus, Enterobacter, Klebsiella, and Pseudomonas, grow within the tree using the sap as a nutrient source. As the sap is used, oxygen in the heartwood is depleted (creating anaerobic conditions), methane is produced, the pH of the sap is increased (pH 6 in healthy trees to pH 7 to 8 in wetwood), and a high pressure develops in the wood (60 psi in affected trees vs. 5-10 psi in wetwood-free trees).
It can be avoided, by minimizing damage to the adult uninfected trees. This will keep bacterial introduction at a minimum. On an adult tree, you will have to live with it, or replace the tree. If you replace it, I'd choose something resistant, like an evergreen, or an oak, or a walnut, anything slow and hard, or coniferous, will be more resistant to the disease. Mulberry, willow, cottonwood, ash, elm, and some maples are more prone Avoid these.
