CC=>State School=>MSF=>Consulting

Spent 2 years at a CC then transferred to a non-target state school. Graduated from the CC with 3.6/3.7 (would've been 3.8 but got a bad grade in one class after being too focused on ec's) and I expect to graduate from the state school with a similar GPA with a degree in Accounting. Due to going to a non-target, I'm concerned about job prospects, though with a 3.5+ GPA in my major I'll easily land in Big 4 audit since they recruit constantly at my school, but it's not what I'm looking for. If I don't get a good job offer, I'm considering going straight to an MSF program at a Tier 1/2 school in Europe after undergrad like IE, RSM, LSE so I can relocate easier and have better job prospects. Not particularly interested in staying or returning to the US. I'd be aiming for a 700+ on the GMAT and will probably take it Sept/November of my senior year.

My goal is to break into consulting. Not interested in IB. I also would like an MBA eventually, if necessary. The dream would be INSEAD or Oxbridge.

My question is how much will my CC background hinder me if I apply, as I'd be applying with only 1-1/2 years at a four-year school and no WE except an internship or two? I know MSFs are often geared towards recent grads, but with me going to a CC, would I need a year or two of WE to make me a better applicant?