Oh that's a toughie!

Things to consider asking that could potentially sway them your way, besides price:

1. Settlement terms, how long are they after, short or long.
2. How much deposit would they accept, ie - 5%, 10%

You could try asking about the price, see if they would accept anything below the given range. Or if they want mid range, or the top end of it. Personally, if it was me, and it was the perfect house, and I know the other offer is genuine, I wouldnt go with an offer lower than the range.

Good luck hun