Why not scale down the LHC and build something that is capable of accelerating something relatively small say 10-100kg fast enough to make it to orbit instead of accelerating atoms to nearly the speed of light.
The problem with conventional rockets is you need to carry the fuel to get in to orbit as well as the fuel to go where you need to. The bigger the ship the more fuel you need to carry to overcome the weight of the fuel.
If you can split the carrying of fuel for your journey from getting your rocket in to orbit you would not need to waste as much fuel lifting itself.
You could set up an automated system that would fire a 10kg payload of fuel every 10 minutes and get what you need over time far cheaper than one big launch.