Is it possible to finish the Bank Heist without the wire cutters

Solution 1:

Yes you can.

There is some loot you can't get to without taking damage if you don't have the wire cutters but you can reach the level goal without them.

After some time away from the game I saw things from a different perspective and figured out how to get to the goal a different way.

If you are stuck and can't figure it out:

You flip one of the switches, climb up the wall that the switch is on, walk over the cage careful to avoid the trap pressure plates up top and then drop down next to the other switch and flip it, avoiding the pressure plates that reset the switches. If you are in focus mode, the trap plates glow red and the ones that reset the switches glow blue. You need to be relatively quick but not nearly as perfect as the route under the stairs that I had previously been trying.