For the Hulse-Taylor (PSR B1913+16) binary, the orbital period is 7.75 (Earth) hours, so I will admit, you've got a solid argument against one of the reasons I gave as a justification that GWs will be hard to detect.
https://en.wikipedia.org/wiki/PSR_B1913+16
And since we have no information about whether the pulsar is tidally locked (a likely scenario) to whatever else is going on inside of the binary pair, detecting a faster oscillation would not necessarily rule out the idea that you have detected a GW originating from there. One or both masses could be rotating in other complex ways to produce GWs. Possibly this one of the best reasons for doing the current experiment, although no one here has yet mentioned it. It might tell us something about what is going on INSIDE of the EH of a black hole, something that is not supposed to be possible.
There is no guarantee that this object will necessarily be the brightest object in the GW sky however. It's a very big sky, and we have no clue from visual observation what sorts of processes may be going on up there. So if the period of whatever GWs are detected are different from 7.75 hours, then what? It could take longer to identify the source of the GWs through optical or other means than it has already taken us to construct and test a more sensitive detector.
And as for "How can I not be aware of that?":
Orbiting bodies are only one process that might produce GWs. We know next to nothing about whether other processes like stellar convection:
http://www.astro.utu.fi/~cflynn/Stars/l9.html
could produce GWs as well. The masses which move interior to our own Sun account for the equivalent of thousands of Earth masses moving in small orbits in the interior of a massive body much closer to us than the Hulse-Taylor binary. And as far as I am aware, even GR models of gravity don't tell us anything about gravitational-thermal processes moving masses interior to a star. But if they did, we could predict things like solar mass ejections, couldn't we? That sounds like a more than marginally useful thing to investigate.
For that matter, is LIGO as it has been configured now or during the last 10 years even been capable of detecting the GWs of the Earth-Moon system responsible for Earth's tides? That information can't (or shouldn't) be a deep, dark secret, should it? Unless it actually can't. The LHC can detect Earth's tides, and it wasn't even designed to do that.
How can YOU not be aware of THAT? Don't you KNOW anything about the inverse square law? Or is that another law that applies to all other kinds of waves except for GWs? I kind of expect that if a local interferometer based GW detector would work for detecting much closer sources, we would already have seen lots of GWs using a less sensitive detector a very long time ago.
So, now as far as I am concerned, there are several more reasons to expect this experiment will come up with another null, or at most a perplexing set of results.
And THAT (not knowing) is the principle reason we do ANYTHING in science in the first place.