Is your prob hooping, or stitching thru the insulation? You might want to make an applique patch and sew or glue in place.

P.S. I don't like that you've used your email address as "user name"...afraid many spams might come your way. Can you change it?

I have seen this done but the owner deconstructed the bag to expose the outer fabric and kept the insulation pulled back away from the embroidery. If you stitch through all layers you won't have an insulated bag. That inner layer is not strong and all the stitching will weaken it plus the bobbin thread will be seen and can get messy.

The best option if your heart is set on the project is to rip out the stitching on the sides of the bag enough to hoop your fabric. After your embroidery is done, restitch the sides. It can be intense and frustrating. If you have an old lunch bag, use that as a prototype. Maybe you can find one at goodwill or another resale shop.

OR

There are patterns for insulated bags (I saw some in the Kwik sew book yesterday) and places like Joanns have the insulated "fabric" that you would use for the liner. There are other insulation options like a thin but stiff foam that is really nice for bag construction.

Here is a tutorial on making and embroidering a lunch bag. I have done several other projects from this site and all have worked out well. You might be able to reverse engineer for the embroidery portion to incorporate that into your bag. The pictures will give you some inspiration.

http://www.emblibrary.com/EL/elprojects/pdf/PR1737.pdf

Good luck and I hope we can see pictures when you are done.