This sounds as though the aberration in the laser's output could be fluctuating owing to "mode hopping" (where several of the laser's cavity modes are active and playing a time varying role) so that, even at a constant output power, the aberration of the output beam varies with time. Wavefront aberration is roughly the Fourier-dual of Strehl ratio. This quantity in turn is roughly the same as the coupling co-efficient of the wave into the single mode of a fiber, with equality holding if the fiber's mode field diameter and mode shape match the diameter of the Airy disk for the focussing lens.
In the past I have used the following effective technique to check for mode hopping ......... : couple the laser output into a single mode fiber, measure the laser's output power and watch for fiber output intensity fluctuations further to fluctuations in the power! In other words, I've used your exact setup! So this isn't going to help you much because this is what you're seeing and you need some independent way to check that the phenomenon is what I am saying it is.
You could try focussing through a pinhole with the same diameter as that of the Airy disk for the focussing lens and check whether the "breathing" still happens. This would be a good sanity check, but it depends on experimental kit that is less mechanically stable than a fiber mounted on lens, so you'll need to be patient and quiet to check whether this setup gives the same results as your fiber. Otherwise, the pinhole output will vary even more wildly owing to mechanical vibration.
I would collimate the beam, or have it very slightly diverging, so that you can look at the farfield pattern on a wall (take due heed of laser safety considerations here); try imaging it through different polarizers so that mode hops may show up as farfield intensity variations. But this is not failsafe.
The only sure fire way to detect mode hopping aside from with your setup is to input the laser into a point diffraction interferometer or other interferometer that makes its own reference (by passing one copy of the beam through a subresolvable pinhole to strip off any wavefront information and using this to build the interferogram). A wavefront sensor may achieve the same thing, but in my experience these will give you an accuracy of about 0.07 waves RMS. Much smaller aberration changes than this will lead to a well detectable difference to the coupling into a single mode fiber.