MBA Decision: Stern (full-ride) or Kellogg for finance?

I'd urge you to do two things:

1. Ask Kellogg's financial aid department for a counter-offer
2. Go to the admit weekends

Like you, I've also been admitted to Kellogg (albeit in R1) as an international student with a finance background with the goal of eventually breaking into PE-style investing. I attended DAK, Kellogg's admit weekend, and have gone through a similar decision process. Here are some thoughts and insights:

• PE recruiting as an international student is extremely tough. The vast majority of US-only shops will not consider you, because they won't risk losing you after a year over visa issues. That leaves you with international shops and those are even more competitive.
• As a consequence, some internationals go abroad in their second year in order to network with shops in geographies where they can actually get an offer. If that's something you consider, the perceived brand of the program in your geography matters.
• Kellogg is underrated for finance. The fact that they place less students in finance than other schools is driven by self-selection and nothing else. At DAK I talked to several students who successfully recruited for PE, asset management and hedge funds. While there aren't many students interested in them, those jobs are as much on the table as they'd be at other M7 schools.
• Kellogg places very well at top BB's/EB's. I'm not too familiar with Stern, but I'm aware it's said to place well in IB. While I have no reason to doubt that, I'd suggest you check out which roles and teams fall into their career report's IB category and where people end up on average.
• Kellogg and Stern tend to attract a very different crowd. Add that to the geographical gap (Evanston vs. Manhattan) and you can easily see why your MBA experience at the two schools will be very different. You may like both, but I'd argue that most people will feel home at one but not the other. That's why in my opinion, it's so important to attend the admit weekends; alternatively, talk to as many students and recent alumni as humanly possible.
• Not sure if that's already factored into your calculations, but cost of living will be much higher in Manhattan vs. Evanston.

I tend to agree with @Pmc2ghy": if you're dead-set on IB, there may not be a big difference in recruiting outcomes (assuming that Stern is as strong in that field as is commonly said). That being said, there are more considerations to take into account. It is up to how you weigh them.