Non-EU citizen working in one EU country and living in another

The title sums up my situation. My wife is an EU citizen and works in Bratislava, Slovakia, and I just started working in Vienna, Austria. My problem is, from every legal and non-profit channel I've been down, I cannot, as an American, work in Austria and live in Slovakia. We want to continue working where we work. Here are the options I have been told:
- make a 'fake' leasing agreement with someone living in Austria though we don't live there (our friends, hard working families who would rather not do anything illegal; the immigration agent said "No one will check". So, it's not 'technically' legal.))
- rent out a small flat or room in Austria (I prefer not to throw away money)
- move across the border. (This is what we're looking at now, just to move right across the border, close enough to Bratislava and still in Austria; oddly enough, flat owners are saying they can't register me if we wanted to, which is obviously an injustice)
- quit the job, stay in Slovakia 5 years till I have citizenship

I know there are a lot of free-minded people out there, but I am not one to do things illegally. The people at the Slovak immigration center did not advise that applying for Slovakian residence was an option; especially since Austria requires residence.

Is there any legal way for us to live in Slovakia and for me to work in Vienna? I mean, really, my wife is Slovak; so, we both are forced to move out of the country just because I found a job in Vienna (thanks to the row between the government and Slovak Academy of Sciences).