Of all plants, bulbs are the easiest to move. The bulb part is a storage facility of moisture and sustenance, so they have a much easier time adjusting than other plants that do not have this feature. There are probably some bulbs that, if disturbed during a certain period, may not make flowers that year, or the following year, depending on the timing of the disturbance, and whether or not they are mature size. In the scheme of the decades of the life of a garden, this is hardly much of a wait compared to growing a perennial from seed, or sprouting an acorn to grow an oak tree. IMO, no bulb that is moved within the course of 1 day is as disturbed as a bulb that's been in a plastic bag, in a box, riding around in a truck, hanging on a shelf. There's no logic in that at all. Bulbs don't do nothing over the winter, or summer, they're busy underground like any other perennial plants.
In early April, I moved thousands of bulbs because we were moving. I knew it would be fine because I've always moved bulbs whenever I wanted/needed to with no ill effects. Constantly moving would be a problem, but a 1-time move is fine for any bulb, especially to rescue it from mowed areas. Occasionally gardeners re-arrange beds, the bulbs don't mind.
I found many bulbs in our mowed areas that have been moved to beds over the years (and then recently ALL moved.) Lycoris, Gladiolus, Watsonias, Oxalis crassipes, Alliums, Daffodils, all moved when found, after being mowed for years. Once you recognize the foliage, you should be able to spot them among the grass, even when not in bloom. (And if you don't move them when you can see the foliage, you can't know precisely where they are and the risk of chopping into them is high, which would be much more of a risk to the bulb than moving it intact.)
The two bulbs you mentioned are often put in lawns because they can be a bit weedy for garden beds, and as you mentioned, usually bloom before the mowing starts. When put in lawns on purpose, people wait to mow until the bulbs bloom. Sounds like the mowing is not under your control though, in which case making a bed specifically for these bulbs should be fun and allow you to see the blooms. If you wanted to do other gardening, a separate bed from the bulbs might be best, but entirely up to you, the gardener in charge at the moment at that property. As short as these bulbs are, I think they would make a fine ground cover under taller plants, but you'd need to keep an extra sharp eye for grass sprouts, which, at a glance, would be nearly identical. "Mower guys" often don't take the step of aiming the mower away from beds, which can deposit grass seeds in beds.
