So for these "back to back" bookings, the hotel should allocate you into the same room for the entire stay, your keys should be programmed to open your door for the entire stay, and they should roll any room charges from booking to booking until you pay on checkout.

One potential "issue" is that hotels often start to offer a discounted rate on the room for bookings over three nights, so you could potentially be paying more per night if you do four individual bookings instead of one longer booking. Obviously, in your case, you can't even make the single booking anyway.

If you do make the individual bookings, I would recommend that you still contact the hotel and provide each of the booking numbers to "inform" them that you are staying for multiple consecutive nights. This way, you avoid the unlikely but potential issue of the hotel not being aware of your extended stay when you arrive.