mIRC does have an option to rejoin channels when kicked in the Options menu.
After ages of searching I finally found that! I'd looked everywhere under all of the subheadings but never clicked on "IRC" itself. I've selected Rejoin Channel When Kicked, Rejoin Channels On Connect and Keep Channels Open. I think that lat one may have been what I needed. We'll see what happens. If the problem goes away I'll start deselecting those one at a time to find which one actually fixed the issue.
