Why don't you want to arch your back? If you're worried about arching your back too much, then wear a belt. But keeping your back flat kills your power. Arching your back will give you better leverage. If your bench is too tall, then put a plate or two under your feet on the floor. You will probably be okay as long as you keep your butt on the bench...no one I know has hurt their lower back unless their butt was way off the bench.

If you do put more than one plate down, then be careful because they can slide. I have a bench at home that is about 95 feet off the ground. I put rubber mats under it and I have some shallow wooden boxes that I put my feet on.

There is nothing specifically wrong with keeping your feet on the bench although it makes it difficult IMO and feels unnatural.