That concern is exactly why the U.S. Small Business Administration exists. The federal government actively works to award ~23% of prime federal contracts to small businesses each year. Moreover, that quota contains specific goals for awarding contracts to Women-Owned Small Businesses (WOSB), Service-Disabled Veteran-Owned (SDVO) small businesses, other "small disadvantaged businesses," and businesses in "Historically Underutilized Business Zones" (HUBZones).
If you actually try to "undercut these large, wasteful, stupid and taxpayer moneysucking behemoths," you'll have federal policy at your back.
Furthermore, "[f]or all procurement actions expected to exceed the $150,000 simplified acquisition threshold, prime contractors are required to make a "best effort' attempt to make use of small disadvantaged businesses, SDVOs, and WOSBs as subcontractors if the opportunity exists under the contract. For procurement actions expected to exceed $650,000 ($1.5 million for construction), the winning contractor is required to provide the agency contracting officer with a written plan that establishes a small business subcontracting goal. The plan details how the winning contractor will make use of small business in each subcontract category and provide for timely payments." [0]
[0]: http://www.sba.gov/content/about-government-contracting