Please be sure to let us know how many adults/children you're including in your search as this makes it easier for us to find the same hotel by price as well as ensures that we are researching the same property.
4* HOTWIRE Walt Disney World - Bonnet Creek for $370
==> this does match previous reports for the Beach Club, but you can't necessarily exclude the (not yet reported to the board) Yacht Club as a possibility
I did look further into the double/quad occupancy above...
HOTWIRE used to define 'children' as age 2-17... they have recently changed this on the 'Hot Rate Hotel' side to indicate ages 2-11.
When searching for a regular rate hotel you can still input the number of children and their ages.
On the Hot Rate Hotel (opaque) side of things you can now only select the number of children, not their ages.
==> i'm not sure if Hotwire (opaque) is treating any children as adults because that is the way it is coded for all hotels OR if any children would still be counted as children for hotels where they treat any guest 17 years or younger as a child
More specifically to your dilemma...
Your in a 'grey area'. I know in the past users were able to book Disney properties for double occupancy and then call the hotel to add the children name(s) to the reservation. The other issue (how big/small this is depends upon your outlook at the age of your children) is what if the room only has a King bed... although this may be less of an issue in Orlando since the hotels here are used to accommodating families and therefore have a higher percentage of rooms with 2 beds in them and/or enough space for a rollaway assuming there is no pullout sofabed (vs. if you were staying in NYC or Chicago where hotels have a high percentage of rooms with only one bed and little/no room for a rollaway).
Given the price difference i would be more inclined to only include the 2adults in my search and then call the hotel to add the child(ren), as well as sign-up for FastPass and any other perks. If there is an issue adding the children then i would offer to pay to add them to the reservation (since you would have paid Hotwire anyway). Since you are also considering splitting this stay into two reservations, you could make a first reservation for just one nite (or the first few nites), see how it goes after you call the hotel to add the chidlren, and then use that as a baseline to decide if you book with the remaining nites as just 2adults or if you include the children on the original reservation.
NOTE if you do go the split reservation route (and are hoping to get the same hotel for all nites) before any purchases you should first check for each set of dates you intend on booking to make sure the same hotel appears for both searches.
Please remember to return to BetterBidding and use the PRICELINE and HOTWIRE links on the board to begin your travel purchases... or thru our sister site MAPSandHOTELS.com for your regular rate purchases.
Let us know how you'd like to proceed, or how it goes should you move forward, and we'll take it from there.