Ah, I always forget about satin stitching applique since I rarely do that. I really, really hate to suggest that the only option I can think of to "fix" it is to start all over, but in this case, I can't figure out how to get the puckers out.
Here's what I'd suggest doing next time:
1) Heavily starch all fabrics before starting (especially the applique pieces)
2) Get some WonderUnder fusible web so your applique pieces don't shift while stitching.
Apply the fusible to the back of the cat & window it. Peel of fusible to uncover the part of the cat that is over the chaise. Fuse the cat to the chaise & stitch the cat down (yes, you will also need a stabilizer -- a tear away paper stabilizer has worked well for me with satin stitching. You definitely want something you can remove/dissolve so you're not adding extra layers to stitch through later on. Once the cat is stitched to the chaise, add fusible web to the back of the chaise & fuse the chaise/cat head to your background fabric & window it.
3) Stitch around the edges of the cat & chaise and then start your beading right away. Fusible gets stiffer the longer it's had to set. If you're doing the beading by hand, you'll want a sharp new beading needle & I use the Poke-a-Dot stick on thimbles so I can grab my needle/beads more easily than with a traditional thimble. If the fusible starts to get hard, you can reheat it to make it warm & pliable, but be warned that you can only do that maybe a couple of times before it stops working. If you want to avoid the aggravation, PaperPrincess is right that hot set beads are the way to go!
4) Once you have the beading done, you can add in the detail stitching & your tufts. Personally, I like doing both of those by hand because it gives me more control over the project. You could either do the detail stitching through the batting or you could just do it through the fabric. It just depends how puffy you want the finished look to be. Tartan gave you excellent information about doing trapunto. I'd either use a high-loft cotton like W&P or a wool batting so you can really see the definition of the stitching/tufts.
5) To make the chaise/cat really stand out, you can do a small fill around the design once you've layered everything for your final quilt & it will puff up of the quilt and look like you stuffed it.
A couple other notes:
* I'm not sure you have a wide enough seam allowance on the left side. If it's not at least 3/8" (maybe even 1/2"), I would move it slightly to the right so you don't get the trapunto batting caught in your seam.
* I would add the legs on to the chaise when you're cutting. You don't need to put the trapunto batting under them, but it will be so much easier to stitch around everything at once versus trying to add some legs on later. You'll just want to add the detail stitching at the top of the legs when you're doing the trapunto.
AND FINALLY ... after looking at all these steps if you would rather just be done with this thing & move on to something else, one option is to cut a small slit in your background fabric underneath the chaise (be very careful) & stuff it. I've done that with lettering to make a child's name puffy. You can fill it with wool yarn or fiber fill. The only thing you may still have to do is to pull out that one line of detail stitching (1/4" above the satin stitching line at bottom of chaise) & redo that. You might do better sewing it by hand since that gives you more control with spreading out the puckers. If you stuff it enough, the seams will be mostly hidden by the fabric. The areas with the puckers will likely look slightly different, but the difference won't be nearly as noticeable once it's stuffed. You definitely want to do your beading before stuffing the top area or the distortion will make it even more difficult to stitch through everything.
Good luck! It's a really adorable design & I can't wait to see your finished product!!!!!