You have an understandable misconception which is clouding the issue for you. The "circular" in "circular polarization" doesn't refer to the way the light is polarized in a different plane as you rotate the filter. It refers to the way the waves of light themselves are aligned. Check out this (public domain) illustration from Wikipedia:
Trippy, huh? But really, this part doesn't matter. The problem is that linearly polarized light might interfere with metering in an SLR camera. That's because the half-mirror used to direct light to an SLRs' metering and autofocus system also has a polarizing effect. The concern is that certain alignments of incoming linearly-polarized light might get canceled out, affecting the metering and focus. Using circularly-polarized light bypasses this.
But, really, it's not this wacky radial polarization that's useful. It's the linear effect that's really desired — that's what keeps out light that's aligned inconveniently in your scene. So, why does a circular polarizer work anyway? Another picture (again, public domain from Wikipedia) makes this clear:
You can see that the "normal" light coming in from the right side is first restricted to being linearly polarized, and only after transformed into the convenient-for-metering circularly-polarized light.
That was long, so here's the summary answer in bold: In either a simple linear polarizer or a more expensive and complicated circular one, it's rotating the orientation of this linear polarization that gives the important effect.
Your point & shoot camera doesn't use a half-transparent mirror to direct light to a dedicated set of metering and autofocus sensors, so there's no need to do an additional transformation to prevent interference. So you can just go ahead with the linear polarizer and it will work the same way.
(All of this said, I've seen pretty good anecdotal evidence that linear polarizers don't significantly interfere with metering or focus in most SLRs either. Presumably the polarization effect of the half-mirror isn't as strong as assumed by the circular-is-necessary theory.)