Below is a pretty good definition of a soft shell jacket - imo I think of soft shell jackets as anything not waterproof like with a Goretex finish which could (1) be damaged by heat, and/or (2) resist application of the transfer. Most important for you, though, is that they are almost always made of synthetics which can melt from high heat (i.e., from ironing).
Your best bet would be a cotton/synthetic blend, no high-performance finishes, and be sure to test with an iron on a place inside the jacket that no one will see. Good Luck!
Soft-shell Jackets are designed for use outdoors as part of the layering system, and can be used alone, ideal for aerobic activity in light weather conditions if they are water repellent and highly breathable, or as an insulating layer under a 'hard shell' outer jacket.
Guide To Soft-shell Jackets - GO Outdoors
The 741 customer reviews & 202 answered questions in the product information link you provide in your question for Avery iron-on transfer papers have a wealth of information and tips for using (or not using) this product. I suggest reading all of them if you haven't already.
As mentioned in the first posting of my answer, ironing temperature is a critical element to success and requires some experimenting with the heat settings on your iron and the exact fabric/fiber type of the jacket.
My experience with these type of transfers is with kids'/teens' tee shirts, in 100% cotton or "cotton rich" blends which means mostly cotton with some low percent if polyester (under 50%), or stretch fibers (under 10%), like Lycra.
Kids' tee shirts are not a big initial investment and they usually only need to get through one party or school event, and the result doesn't need to be perfect. This is not the case with a nice jacket, so the stakes are higher for you.
The potential issues I see with your project are:
You are not sure what the fiber and fabric type of the jacket will be
You won't have extra fabric to test the transfer papers on
The jacket will likely be an investment and it could be ruined
I know you like the idea of the transfer papers, and they are cool. You might want to buy some fabric and play around with the papers and the process before working on a jacket.
If you aren't able to access or use a professional apparel printer who can upload your design and print the jacket for you, you could also look into professional embroidering businesses in your area. Commercial embroiderers can also use your uploaded design file. A complication with these options is that you usually need to print/embroider a certain number/lot of items to make this financially feasible.
One option for you that is less costly and will still let you use the transfer papers, is to purchase some fabric at a fabric store and print on that. Then cut the printed image out and attach it to your jacket by sewing or gluing with fabric glue. If you try this, it would be best to buy your background fabric in the same color as the jacket, and made of approximately the same fiber content and fabric construction.
At a minimum, you could buy one package of the transfer papers and practice on clothing you already have that you can either live without, or live with if the printing isn't perfect.