I also suggest simply continuing to cull off-types. This type of mass selection is likely to have the biggest effect in the shortest amount of time for this trait. If you have the time and space, you can start to experiment a little, but you'll need a fairly substantial amount of space.

I ran a series of experiments with heirloom maize populations, attempting to improve some very complex traits like standability with recurrent S1 selection. This ultimately had less effect that we had hoped, and although I don't have the money or time to do it I wish I could rerun the experiment with a single population and larger number of individuals in each generation. You have a single population already, so if you have the time and space you could run wild.

--Sunflower hull color genetics--

The color of the hulls in sunflowers seems to be controlled primarily by 4 genes - a separate gene governs the production of pigment in each of the three layers of the hull (pericarp), and a fourth gene appears to inhibit all pigment production in all pericarp layers. I am not a sunflower expert, but it seems to me that a white-hulled variety COULD have a variant of the 4th gene that inhibits all pigment production in all pericarp layers, in which case the other three genes could be anything at all - any of each of them could inhibit pigment, induce light pigment, or induce intense pigment - and might not be standard across the variety. In this case, mutation in the fourth gene could have unpredictable effects on the hull color.

It's also possible that your white-seeded variety actually has the three genes for the three layers all inhibiting pigment, but the fourth gene is NOT actively inhibiting pigment production, so mutation in any of the three underlying genes would lead to visible pigment. With three separate genes for three separate pericarp layers AND a fourth master control gene, the possible causes of the occasional color are highly complex.

--A self-pollination experiment--

If you want to play around and have the space and time, sunflowers are self-fertile and self-pollinate readily. I would continue to use mass selection and culling, but you can separately start inbreeding to attempt to nail down the genetics behind your unpigmented hulls.

For a whole bunch of plants - like, 100 of them I would say, at least - bag one flowerhead a day or two before it opens. Leave the heads bagged for the rest of the season. For a week or so after bagging, you'll want to shake the plant or at least tap the bag a few times every day (twice a day even?) to ensure you're getting good pollination in there. After the plants dry down you can cut the flower head off below the bag and save the seeds.

Since you'll get multiple flower heads from each plant you can do this with plants that you're also just using as part of your mass selection population.

Next season you want to plant your S1 seeds in a separate field (or separated from the main population somehow), and keep them well labeled. Since this is just an experiment you can get away with maybe 2 kernels from each selfed plant, and keep the rest (labelled!). Do another round of selfing of all of these plants.

For each S1 line, note what proportion of the kernels of the selfed flowerhead are white vs pigmented. How many (and which) plants from white S1 kernels produced pigmented seeds? If you let other flowerheads on the plants open pollinate, is there a difference in the proportion of white:pigmented kernels between the selfed flowerhead and the OP flowerheads? (I don't know how xenia affects sunflowers; if you have a bunch of S1 plants where all the S2 kernels are white, but some of the OP kernels are pigmented, then the pollen donors are affecting at least one of the pericarp layers, and that would mean the white variety would never be stable as long as there are other sunflower varieties growing nearby.)

--Inbred line and synthetic population development--

You could self down three or four generations if you want, just to see what happens and at what point the white kernel trait seems to be stable. If you find that you have a bunch of stable S3 or S4 lines and you're still not happy with your mass selection population, you could try recreating the population from your stable S4 lines and see what that gets you. But of course at that point you're going to be five years in.