maybe it's because she knows that her other sisters are pregnant and therefore 'ratty and tired' and won't want to organise the BBQ? If she's serious about it, I would take her up on her offer and just turn up!
I would be pissed off too (but am also pregnant!). However I would be taking this as the perfect opportunity to let her do it all herself and run around like a headless chicken, while you and your sister enjoy watching her flap. At the end of the day your Mum and Dad will love having you all around for a change.
I'm not surprised it's made you all huffy, maybe you could say something like, "Wow, that's so sweet! Are you sure you want to? You don't usually get involved in the organising of family events, it's a lot of work, you know."
Can you not just let her? Hand over your fiver and do nothing else -it's her problem now. Or say 'how nice, we were planning the same. Great minds and all that. Let's do it together'. I would just hand over the fiver and wait for it to go tits up but I am VERY grumpy at the moment too and do not advise this. I am a Bad Person.
OTOH though it means she's pulling her weight for a change, which is good. In your position, being completely honest, I'd probably just bitch about her with other sister and on MN but let her get on with it. That way you get your bitching, she gets to play Waltons, you don't have to be all organisey for a change. Everyone wins
Then I would conspire with your other sister and text/email - something along the lines of..
'Thank you SO much for organising this party - it is so nice for you to take a turn!! Mum and Dad are SO looking forward to it! It is SO nice of you to step up to the plate. Both Sis1 and I are really busy with kids/work and of course its tough being pregnant, so we want to just let you have the freedom to get on with it yourself. It will be a real pleasure to just sit back and enjoy a family gathering with us all together. Thank you SO much again!! XXXXXXX'