Hi,
I'm not sure I understood everything.
The node name is purely visual and should not be used to identify a node in the code. This is the job of the id.
It's possible to increment but only if you have developed the node. Otherwise you have to add an "nodes:add" event which when you add this node, updates the name by modifying the node properties.
RED.events.on("nodes:add", (node) => {
// Skip if not your node
if (node.type !== "your-type") return;
// check if name exist with
// RED.nodes.eachNode
// then update it
node.name = "new name"
});