I personally have not tried wire with bobbin lace but I do know people that have. From memory, they used all wire & no threads. I do know they had special bobbins for the wire, but that is about all I know. Just experiment, see what happens, & remember to let us know how it goes.
I believe the bobbins used for wire lace are called "hookies". I saw some recently on the "lacemaking circle" website. As an alternative you can wind a single layer of thread onto a regular bobbin, and then wind the wire on top of that. The problem is that wire slips when it touches wood. You need something to stop it slipping as you work, so either the hook or the layer of thread. There was a website called silverpin studio run by a woman who does a lot of wire lace. She might be able to help.