Doing exactly what you said (not lying about your preferences) is what you need to do. You need to explain your circumstances to your potential employer, and if the employer is not flexible enough, they will say three things:
1. We'll wait on you to make a decision.
2. Contact us when you know more, and we may or may not have a spot for you.
3. We have to rescind our offer.
Hopefully they won't say the latter, but they might. Again, don't commit to anything that may burn bridges with employers at a later date. Likewise, don't pretend that lack of funding isn't an option (which you don't seem to be doing
). Just be completely honest, and things will work out for the best.
Edit: My "backup" for not getting into a top program is staying at my current institution and completing a masters in mathematics with a thesis.