Any chance you can find a duplex to defer the mortgage payment to a certain degree? The price might be higher but with the rental income, the overall payment might actually be less for you.
Another option is to find a hard money lender to partner with to make a full-cash offer. It'll cost you a little more for the hard money up front but as other's have mentioned, you might be able to offer less and get the place with a full-cash offer. Then after the purchase, you could (re)finance the home and pay the hard money lender off. There are a number of things that could go wrong here (doesn't fit into the loan specs that you want, the house doesn't appraise at what you need it to appraise at, the hard money lender requires more equity in the deal, etc.) but I'm trying to think outside the box to make your initial offer appear more robust.
Another option is networking with as many wholesalers as possible at your local real estate networking events (REIA) and try to find something that can pass for an FHA loan or whatever. I've seen a few pretty solid places come through my email that might be able to pass for a conventional or FHA loan. I know some stud and studette wholesalers see houses of all conditions come across their desks. With that said, most of them require quick closes and cash offers, but I think they get long runways every now and then. You may have to pay more (once again) but you might be able to get something without having to claw it out of the competition's hands.
I'm struggling to put my thoughts together on my next idea so I'm hoping that others can fill in the details. What types of loans can you get (conventional?) where they can wrap the rehab costs into the loan? This might work in the situation that you buy a fixer-upper duplex, live in half and rehab half. Then flip to the other half. I hear of people buying fixer-uppers all the time and getting rehab costs wrapped into the loan.
At the end of the day, don't over-extend yourself. I know starter homes in the Twin Cities are getting harder and harder to find. That also means that it's starting to make sense to rent rather than buy. Maybe you should rent a primary residence and buy a rental prop in another market. This could help you offset your current rent. With your high FICO score, and a W2 income, you'd probably pretty easily qualify for another market where there is a little less competition and better rent to value.
Okay, enough thoughts for one post. DM me if you want me to elaborate.