OP, it occurred to me this morning you might be able to use the surface mount plastic channel (see Lowe's, ACE, or Home Depot). This product comes in various sizes, and has components to go around corners. The peel and stick tape will probably not work on stucco, but a couple of concrete screws will easily hold this up over the short distances you are describing. The wide version (about 1 to 1.25 inches) is wide enough to mount the new door bell button, and should be wide enough to cover the hole at the old location, and there is enough space for splicing the wires. Also, most of these products can be painted to match your house. Send me a PM if you would like to discuss further. This would probably cost less than $10 or $15 for material.
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Pennsylvania, for 60+ years, most recently, Allentown, now TV. 
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