Hello, I'm sorry to hear about Elsie's bad habit of eating whatever she finds and I understand your concern for her as if she eats the wrong thing she may get very sick, either from an intestinal blockage or a toxicity.
Large amounts of fat in the abdomen can feel firm but it is also possible that she has some organ enlargement which is making her abdomen feel firm.
Does she drink a lot of water as well?
Because she has a voracious appetite and weight gain it is possible that she has an underlying disease process such hypothyroidism or hyperadrenocorticism. Dogs with poor thyroid gland function will gain weight, are more likely to have poor skin and coat condition, and are often lethargic. Dogs with adrenal gland disease will often have a very good appetite and increased water consumption, abdominal enlargement due to organ enlargement, poor muscle tone and increased abdominal fat, they may also have poor hair coats and easily get secondary infections.
I do recommend that she see her veterinarian for an examination and some blood testing to make sure that there isn't anything underlying her behavior and physical appearance. I would start with a normal complete blood count and biochemistry profile as well as a thyroid profile. Depending upon those results further testing for hyperadrenocorticism (Cushings disease) may be indicated.
Unfortunately if she has a medical problem we probably won't be able to change her scavaging behavior much until that it is under control. Your best bet now is going out with her every time she goes out so you can stop her from eating what she shouldn't. It may help to give her a special toy to carry only when she goes outside. That will slow her down and when she drops the toy you know she is ready to pick something up and you can stop her.
Best of luck with your little one, please let me know if you have any further questions.