Loss of precision due to low-pass filtering in physical units
In general, no.  In fact, if the low-pass filter characteristics were well-chosen in the sense of getting the best post-filtering signal to noise ratio, then the amount that you might degrade any such measurements would be minimal.

In part this conclusion comes from my "well-chosen" qualifier.  Roughly speaking (and I can only speak roughly because your problem is so vaguely stated), if you've got a true feature in the object that you're imaging that resolves to an image feature as sharp as your 1-pixel line then unless you have absolutely tremendous noise your $\sigma = 5$ Gaussian filter is poorly chosen.

I would assume that for looking at tissues, you're looking for features that are significantly larger than 0.05mm, and in x-rays you're getting significant white noise -- hence, a filter that smooths out the noise without smoothing out the features that you can actually reasonably expect to be there is only going to make things -- including measurement precision -- better, or at least no worse.

(But if I had a shop accident and had some AWG 36 gauge copper wire embedded in me, then I would presume that your $\sigma = 0.25\mathrm{mm}$ filter may blur the edges too much -- but that's a different problem, and thank goodness that copper wire shows up very strongly on an x-ray).

In general, what you can do is to set out the problem with more specifics: i.e., what's a typical feature you're looking at, how do its edges naturally merge into the surroundings, what's its contrast in the image, and how much noise is there in the image.  With all of these in hand, then you can start asking how much a given filter will impact tasks such as identifying the location of an edge (or, since the subject is mammography, presumably identifying the size of a lesion).