Your bags should be checked through, so bring minimal amount of stuff to use a hotel and take your pick. I'm not sure if you can stay airside past the last flight, never tried, I don't know if you will be forced to hang around the arrivals areas or the like, but to me a cheap room seems the way to go. Unfortunately from T3 at that time you will have a bit of a hike getting to T5 or T4 anyway, so i don't know that you gain much from using one of the connected hotels over just hopping on a bus to get to the Renaissance, say.

The Renaissance is nice, they recently re-did the lobby areas even if the rooms are dated (it was the first LHR hotel). The lounge is right over the runway (as are a good number of the rooms around it) and the views are spectacular (not that you'd get much use for that). There are some cheaper options around there.