No-one likes a new camera and kit more than me. But at a risk of putting my head above the parapet, I think you are at risk of overkill!!
I may be wrong, but I would imagine your images will be used on a website (Which nowadays will be accessed mainly on someone's phone), and printed media knocked up on an office printer. Top notch image quality just isn't an issue, and nor is perfect image framing. They just need to be bright, and show all the relevant features of the room, with wide enough angle without over-distortion.
I am also guessing that time will be an issue, and you will want to be in and out as quickly as possible.
An 18mm lens on a "normal" dSLR probably wont be wide enough.
I don't know the current state of the market (Whether you will get the wide angle you need), but I would start looking at compacts - though I know they seem more sold on extra zoom, rather than wideness. If not, then an entry level DSLR Canon or Nikon with a kit lens, and something around 10-16mm zoom, probably bought second hand.
I wouldn't bother with a flash either, I would just crank up the ISO if it's a bit gloomy.
You could go in with the tripod, the flash, the diffuser, carrying a 5kg bag of kit up to the top floor of the black you are letting a flat in, then spend 5 minutes setting up and taking every photo to perfection, but I would bet money on you only doing it a few times.
Don't all letting agents just use their phones?