The cheapest rate I've ever seen this hotel go for (with taxes & services fees) is about 45,000 JPY for 1 night, which is over 400 US Dollars. Most of the time it's much more than that going between 500 and 700 dollars per night.
If you are willing to shell out that much money for 1 single night in a hotel....then a 1,000 JPY egg charge (which is just over 9 Dollars) should be at the very bottom of your concerns.