I know that you were just laughing about taking your dad out, but I think that you might have hit the nail on the head there.
How about taking your mum out every other week? Then the weeks in-between, as someone suggested, have one week at your mums house (either with, or without your dad there as well) and then one week where you and your dad go out and have some father-daughter bonding? Your dad might appreciate being able to talk to you (without your mother nearby) or just enjoy your company. On days that you and your dad go out, leave your mum with a gift basket.
Gift basket ideas - A film she enjoys, something towards a hobby (if she has any), some nice smellies - for a long relaxing soak, arrange for a hairdressers appointment etc.
Another idea might be to find out if the U3A operates nearby (University of the Third Age - 'More mature' adults meet once a month, listen to talks (on a variety of subjects), arrange days out etc). Since my mum joined (not that I have any problems with her) she goes out for 'lunch club' once a month, has been on the folk train, had her Christmas dinner and is planning a few other things! Maybe your mum needs some outside interests that aren't just you and your dad.
Oh, if you do take your dad out, don't forget to bring you mum back something (either chocolates, a bunch of flowers, a cake or something else). The last thing you'll need is your mum doing an overly dramatic 'dying swan' routine everytime you both leave and come back. You need to make it seem like taking your dad out is doing her a favour (giving her time to relax on her own/watch what she wants on the TV etc).
Just a suggestion, but before you leave to go to your mums (or before you get out of the car), look at your SW book and your shiny sticker. Then look at a picture of your mum and say "Look at what I have achieved, on my own and without your help. You may have helped me get to the weight I am, but I am changing it myself and you will no longer control the direction of my life. I will reach my goal and you will not divert me from my path." Pah! - it may sound corny, and it might not help but I think that just looking at your SW book will motivate you to make better choices.
Another suggestion (I'm full of 'em!). Every hour that you are with your mum, leave for a couple of minutes (either the toilet/paying for something/looking at something in another aisle etc). Take a deep breath, congratulate yourself for managing another hour. If you feel stressed, and 'need' to have something 'naughty', take some mints with you - I don't know how many syns Trebor Extra Strong mints are, but they sure concentrate my mind!!
Rather than taking her out for a whole day, would it be possible to take her out for a couple of half days? You might find that she's easier to handle in smaller doses! Also, if you are the only person she sees (other than your dad), she may 'save up' her complaints for when she sees you. If you saw her more frequently, she would have less to moan about because she had only seen you a couple of days previously.
Sorry for rambling but I hope that ANY suggestions might help.