After considering this a bit more, the initial price should use either your method of the minimum of the most recent and median price, or perhaps even better, the minimum of all the prices in the feed history (currently 3.5 days or 24 hours if this were changed).

While temporarily pumping the price doesn't affect the conversion rate in this model, it does reduce the collateral required, increasing the risk of the blockchain taking a loss (which would be a benefit to the converter), so we should defend against that. Using the minimum of the entire feed increases the collateral required when recent price volatility has been higher, which is probably desired.