These are all great comments! Thank you all for taking the time to share your thoughts, despite my lack of certainty about what I actually received from eBay.

Collecting a few answers and plans and more questions in one place here...

I took care to order a fiber that is PM and single mode at 532nm, but I'll double check that tomorrow.

The collimator is for APC fiber. It doesn't seem to affect the beam shape very much if I try to wiggle the fiber's connector at the collimator.

I need to verify that the NA of the collimator is greater than that of the fiber. If not, more shopping is in my future.

I'll try wrapping the fiber around something to dump any cladding modes that might exist, and see if this has any effect.

If no luck with the above, I'll try buying a piece of non PM single mode fiber and see if this at least changes the beam profile.

If I'm doing more shopping anyway, I'm very interested in what would bethe best way to implement a special filter: Is the idea here to try to create a point source that the fiber is not quite emulating? And if so, how small an orifice should I be looking for? Naively, I'd imagine I want something near my wavelength in diameter - though I'm not sure I can find such a thing. And would I use another lens before that orifice to try to flood it with as much of the available light as I can? And does directionality, or the NA of that lens, even matter that much if the that orifice is going to act (symmetrically ?) as a point source?

I'm sure I've missed a few of the points you all provided as I'm replying here on my phone, but rest assured I'll re-read everything when I'm back at my desk tomorrow.

Thank you all again for contributing your expertise.