You do have to use the Bomb Mod. If it's not working for you, then see if you can upgrade it to the next mod (for bomb mod). It should be powerful enough to break that pillar to bridge across the other map.

You won't be able to do it any other way bro.
And if you don't have the upgrade for it, then play a few rounds of the multi-player grid games to collect exp points so you can upgrade your arsenal. Then come back to the game and you will have all that you collected in multi-player, to help you beat this nuisance of a level.

Goodluck buddy.

I have the bomb mod already on, in my loadouts and a boost from the auxiliary, and I'm completely using any attack possible on the pillar to get it to break but no luck. Is this some kind of glitch or bug, because I've tried looking EVERYWHERE for hints on this but I can't break the damned pillar.

I am on the PC version and have no way of getting past this.

hmm, if it is a glitch, the best advise I can give if any, restart the chapter. >_<
I remember having to select this mod on the main screen, pressing the #2 key. Not necessarily on the load out screen. Loading up on the load out screen doesn't activate the bomb mod disc as far as my experience when I beat this chapter. I had to press the #2 key to get it to select this specific mod, then use it (speaking only if you're using the keyboard as your controls. Otherwise if you are using a microsoft gamepad in which you use the D pad to select the mod manually.). The icon on your disc bar should turn red as an indication you have the bomb mod selected.

If all of the above is as is and it still doesn't work, that sucks. A redo of the chapter may be your only hope.

I tried starting a whole new character, still no luck. Every combo imaginable, I even tried using a trainer to get max level to buy every upgrade, still no luck.