Let me think...

I approach dove's eyes in two different ways, depending on the design.
A single dove's eye surrounded by kloster blocks: I start and finish the stitch by slipping the working thread behind the middle stitch of the kloster block. So, after securing the thread, to start the dove's eye I bring the needle to the front and to the left of the central stitch in the block as close to the cut edge as I dare. I then take the needle under the middle kloster block
stitch so that the working thread is to the right. Then I work the dove's eye in the usual way. To finish off, I bring the working thread under the first thread of the dove's eye (as usual), then bring the needle up through the fabric and slot it behind that middle kloster block stitch, just like you did to start. Now take the working thread straight back through the fabric as tidily as possible and finish off. This give more of an illusion of the thread having looped around all the kloster blocks, and should make it difficult to see where you started or ended the dove's eye. Sounds more complicated than it is!!
I also shift the needleweaving around with a needle and/or my thumb sometimes to make it look better.
I do a dove's eye from a woven bar pretty much the way most instructions say to.