Easy way to do it, weaken one of there forces down to like 8 HP and then surrender (Button on Pause screen) and then next wave send your rushers right at an opening in their defenses really quick and you'll get it
For "Old Fashioned Way" start a New Campaign, then go for one of the starting battles. They are - being tutorial - initially weaker, so you can reliably win there w/o bombs.
That was nice to play that game The best way to complete the quests with(out) bomb is making it at one of the first fights Blitz is like you said, make them lost a lot of health, lose the fight and beat them with sparemans/knights
But I have a problem with the quest "Power hungry" - I used the power 100 times, even in one battle (I have counted that) and the quest isn't still completed. Anybody has the same problem?