I don't have a very good mental image of the bag, but I would take the bag to a fabric store and get fabric that matches or coordinates with the zippers then make a patch (embroidered, painted, otherwise decorated with something that is more meaningful to you) to cover the logo.
In response to the questions about whether the bag is entirely duct tape, the answer is yes, except for the straps. I made sheets of duct tape by laying tape (in the interior color) cut to the right length on a table with the sticky side up, overlapping each piece by about 3/8". Then I used the tape that I wanted the exterior of the bag on top of that, sticky side down. After I had my sheets, I cut them to shape and size. Then I used strips of tape (cut in 4ths lengthwise) to build the bag "seams". The only part of the bag that isn't pure duct tape are the straps, which are build from tape wrapped around rope pieces to build thickness and shape.
I have been loving this pattern for the Evelyn bag by ChrisWDesigns. As soon as I saw this challenge, I knew I wanted to make a version of it in duct tape - perfect for the rainy season when any purse I use gets wet while biking in the rain. I left off some of the pockets (like the elastic ones on the sides) because they wouldn't do well in the rain. There is a zippered pocket on the back of the purse, and it closes with a magnetic snap. I tried to use the black tape to put some details on the flap (a simple line and a bow), but the dark tape did not want to be photographed.
If you are hesitant to cut into the leather, maybe make a test bag out of a less important fabric first to make sure that everything works the way you think it should.
I don't know off the top of my head the 'best' way to sew the pieces up. Without actually playing with the fabric pieces, my instinct would be to cut all the pieces, sew any desired pockets to the lining (or exterior). Sew the front/back to the zipper (with the lining, like you're making a lined zippered pouch, but with that extra triangle build in if you want it). Then straps to the front/back. Then sew the exterior base. Then finish sewing the lining, leave a hole and turn right side out (make sure the zipper is open when you start sewing everything together).
It's been a long day, so I hope that makes a little sense. Let me know if you want clarification (if no one comes along with a better explanation).