Your GPA will help, name of your uni less important. However bear in mind that at all these unis IR is competitive to get into.
Personally I think your undergrad will put you at a disadvantage, unless you can somehow illustrate some application to your intended focus field in your personal statement.
As a non-IR undergrad too, I can tell you that writing a research proposal for Cambridge without the academic background in place is not easy (and I had a semi-advantage in that my background is economics so I could use an IPE angle). You'll need to do a fair amount of reading to familiarize yourself with the literature in the area you are interested in, and particularly so in order to identify a gap in the literature that you can fill. The upside is that if you can do this successfully and write a decent research proposal it would probably help show that you do have the ability to do the course. I'd imagine you could use the essays you write for Oxford to the same end.
You can apply for more than one degree, it shouldn't disadvantage you. Just on a point of order, it is only Oxford and Cambridge that call taught masters MPhils; at LSE you'll be applying for an MSc.
I'm not sure how much in a hurry you are for this or how much time you may have, but it might help your application as well as your essays/proposal if you do an online course or summer school in IR. I'm not sure where in the US you are, but Harvard and Johns Hopkins both offer IR summer schools, and I'm sure other unis there must as well. Online courses are also an option but would take you longer to complete though obviously you can be more flexible with your time than at a summer school.