Have to agree with wylie, the bank doesn't need to know you paid for a valuation and I wouldn't tell them unless you were struggling for finance and if you are struggling for finance you shouldn't be buying at auction IMHO.
Even if you told them you had already ordered an independent valuation that is non-transferable to a lender unless the lender accepts such - which they usually don't even if the valuer is on their panel. So the point about ordering an independent valuation is a red herring as it is pretty much useless for both finance purposes and purchasing purposes as it is not used for either. I personally think people who rely on 'valuations' to try and bargain down a vendor come across as disingenuous and are usually discarded by any savvy real estate agent. It is simply a waste of money and time and it is really only useful for other reasons like working out your CGT cost base or a fair price for a non-arms length transaction.