Well I tried doing a search for splices and perpendicular splices and couldn't find a good image. I'll have to see if I can find a scan or something. Or see if you can find a copy of Des Pawson's Guide to Knots. the basics of it is that the three strands are spliced though the other rope at a right angle. In my vision, all you'd have to do is splice a small length through, then tie it off with a simple Matthew Walker kot, or possibly a more simple stop knot, such as a wall and crown combo. This would use up very little rope, perhaps close to the amount used by the larks head. And it shows a wider degree of rope skills.
However, considering that your using the rope doubled over (not sure how I forgot that part) I'd suggest a bend instead of a splice.
As for which kind of bend, I'd have to look for a bit. One suggestion would be to use a Double Coin knot (Josophine) as a bend. Make the double coin knot in the second rope, slip the first rope through the loops of the double coin knot and then tighten the double coin knot. A stopper knot at the end of the first rope should prevent slipping through the double coin knot on the second. Should use only slightly more rope than the lark's head, but would look nicer. And due to the nature of the loops in the double coin, you can use it to continue length in vitrually any direction compared to the first rope.