She is either deliberately manipulative or has a strong need to be the center of attention, but either way, it would seem this has to show up in other areas, too, right? Is it just a b-day issue, or do you see this behavior all over the place manifesting in other ways? If so, the posters above are absolutely correct - your DH needs to be on board with setting boundaries as far as MIL. You can't change her, you can only change how you react, by setting boundaries and sticking to them, gently, politely, and firmly.
My ex-MIL also insisted on having DS's b-day party at her house and it escalated from there. I learned to say, "No, I'm sorry, that won't work for us, but we can do x instead." Set boundaries politely, but firmly and nip this in the bud now, or she may get pushier and pushier. If MIL gets upset, that's her problem, not yours. But if DH isn't on board, that IS your problem, and that needs attention, IMO.