I use a wand (medium) on 1/4 Dyneema. I suspect you would have no problem splicing 7/64" with the extra small wand, but for the smaller diameters I've had great luck just using a folded piece of stainless MIG wire.
The folded wire is stiff enough to push up the middle of the braid, small enough to not disturb the braid too much, and you can get the fold bent to a gentle enough curve that it doesn't try to hook strands while you are pushing it into the braid.
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