I embroider on marine vinyl to make book covers. This is not as thick/tough as leather. It sounds like your bag is a vinyl (fake leather) also.
The most helpful thing I have found is to use a titanium coated needle. The ones I use are Organ brand DBXK5 PD. I use them in my 12 and 15 needle commercial embroidery machines. You will need to check if these needles will work in your machine. These needles help tremendously with less friction/heat as the needle penetrates the vinyl repeatedly. They just "slide" through the vinyl better.
I use a medium weight cutaway on the back of the vinyl. I found if I don't use any backing the vinyl can pucker where the design is more dense. I do not recommend using tear away for this reason.
I use fast frames now when embroidering on vinyl. But I used my tubular hoops before I got the fast frames. I won't go back to doing it that way though! The fast frames are the way to go since hooping the vinyl can be difficult due to the thickness. I use the edge of the fast frame to line up the edge of the vinyl and never have to worry if the design is placed correctly. I mark my center before putting it on the machine, line up my mark and go. But I'm working with a flat piece of vinyl (I sew the book cover together after embroidering). Hooping your item may be more challenging!
All of this works well for me. I hope you can use something to help with your needs!
Alyssa Lehmann
www.MissalCovers.com
https://www.etsy.com/listing/245642319/missal-cover-breviary-cover-bible-or?ref=shop_home_active_1