I think the "accrued season" definition only applies to whether you are a restricted (3 or less seasons) or unrestricted free agent (4 or more seasons) when your contract expires. Bell would be a UFA anyway.
What they are probably fighting over is whether being franchise tagged but not signing it counts as the second franchise tag or not, which affects their ability to do it again next year.
I also learned today that the tranny tag does not mean you get zero compensation if the free agent signs with someone else; you get comp picks like any other free agent. You just don't get a guaranteed pick from the tag itself. So there's really no reason not to do it and keep the right to match offers, unless we are really just done with the guy, which is what I would think.