Talking to your current professors is a great idea. GW recruits well (although not as well as Georgetown) for consulting. Have friends with kids that recently graduated (past 5 yrs) from GW who are working in consulting with solid firms in DC.
Maybe more than the school, you should consider what type of consulting grabs your interest. Lots of major firms have federal practices in the DC area. Quite different than their commercial practices (income, work / life balance, travel, types of projects, length of projects, exit ops, etc.) And…some firms will only recruit for federal practice from GW. Doesn’t mean you can’t get to commercial from there but different OCR. Your professors and career services folks should be able to help you.
GW + 3-5 yrs of great work at a great firm should make you quite competitive at top B Schools. People get to HBS with UG degrees from state schools. It’s about the work you do, your grades, your scores, LORs, etc.
I wouldn’t transfer just because of recruiting potential as the “right rocks” find a way of getting that job regardless.