Sam - totally understand the sentiment, but keep in mind that this is not a standard practice with apartment rentals. In fact, some agencies (like vacationinparis.com) will send you the keys in advance so you receive them prior to departure. Also have a look at www.kudeta-home.com for some of their studios, several of which look to be at or close to your budget. Would be worth asking if accommodation could be made for the kids.
As discussed on the previous thread, in your situation an apartment may actually work out to be the better deal when you factor in the ability to make your own meals, along with the comfort factor. Hotels in Paris, especially but not only at the lower end of the price spectrum, tend to have small rooms that could not accommodate your group of 2 adults plus 2 children in one room. In many, there is enough room for the bed with very little extra space unless it is a family room which costs more than a standard double.