I started a new TT position last summer, coming in as an advanced assistant prof. I'm planning on applying for tenure in the fall of 2017, and I've been assured that it should be a "slam dunk." I would apply this fall, but I don't have much teaching experience (1 course, great student evaluations). I do have a lot of experience mentoring students in research, which counts towards teaching at my current school. I'm generally pretty happy, although there are some things about the university I don't love (and I don't see changing), and the cost of living is ridiculous.
This past week I saw an ad for an advanced assistant/associate prof specifically in my research area, at a great school, in a city I would be very happy in. Do you think if I applied to this school they would consider bringing me in at Associate? with tenure? Could I use a potential offer to push for tenure early at my current school? Even without much teaching experience? Both universities at R1.