What auction strategy can get me the item with the least rupees spent?

Fill the bar about half full at the start, then wait. The guy will give you a final warning 10 seconds before the auction ends (I believe he literally says something about 10 seconds). Mash like mad at this point, and generally they'll never manage to get a bid in after you. Once you've got the timing down you can easily get the bid in with 3~ seconds left, and I've not once had someone slip in a bid after that time. If there's less than 5 seconds left there's zero need to stun them, just add a single rupee.

If the auction bar accidentally fills early, do NOT place a bid. Attempt to pay the current bid and your bid will be rejected and your bar starts half full instead of empty. Unless you make this mistake extremely close to the end, you'll have plenty of time to mash A to fill the bar again. Also note that bringing up the bid UI will show you the exact time remaining, and note that the infuriating dialog of other auctioneers does not eat into the time.

You can stun the other auctioneers with a significantly higher bid too (not sure how much, 50 rupees?) and prevent them from bidding, but in my experience if you do this too early it's useless, and if you do this too late you might as well just wait for the 10 second warning and you should pay the same or less as you would have by bumping the price up considerably.

According to recent speedrunners of TLOZ: The Wind Waker, the bidding works as follows:

• If the price is below 20 rupees, no amount of rupees can stun the crowd.
• If the price is 20 rupees or greater, a bid 50% higher than the current bid will stun the crowd.
• If the price is 40 rupees or greater, a bid 10% higher than the current bid will stun the crowd.

Stunning the crowd is almost always the most effective way to keep the price down, as the NPC's usually bid 3 each time and usually happens 2-4 times in the amount of time that a stun would have lasted for. Bidding enough to trigger the stun ends up costing less than the raised price would. Once the price is above 40 rupees, make sure you mash the meter almost all the way to the end to prevent bids completely from the townsfolk.