From an interview with the production designer Nathan Crowley (here), it's mentioned that the Tumbler was partially inspired by random toys from Toys R Us (bizarre, I know).
Crowley told Collider that:
“one of the biggest things [Nolan] wanted to try and do was redesign the Batmobile.” And so off Crowley went to his local toy store.
“It was a Friday, I went away and I went down to Toys R Us and bought a bunch of vehicles,” Crowley explained. “I went over to my workshop and cut everything up and smashed them up. We talked about mashing up a Lamborghini with a Humvee, and so I thought, ‘Well let’s just start somewhere.’ So I smashed this thing together and I left it on his doorstep on a Monday morning.”
So it looks like the Tumbler had no background/inspiration from the Bat-verse at all. It was just a mashup of different toys to make a Frankenstein version of a car.