Transient suppression for on-load tap-changing of 120 VAC ~10 A transformer

[RODALCO]

If you look at the design of commercially-available electromechanical on-load tap-changers, it would seem that what I really ought to do is replace each of my relays with no fewer than four separate relays, to sequentially 1) Connect the mains power (in parallel) to the same input tap that it's already powering, but via a resistor, 2) Disconnect the direct connection between mains power and the first input tap, 3) Connect the mains power to the other input tap via a resistor (thus briefly shorting the transformer winding through the resistors), 4) Disconnect the connection made in step #1, 5) Connect the mains power to the other input tap directly, and 6) Disconnect the connection made in step #3. But I figure there ought to be a solid-state means of mitigating the transient voltage spike without resorting to that level of complexity.

Spot on, you need to look at a TCOL system which is used in electricity substations who work on that principle.
You don't want the loaded transformer going open circuit but by means of a hi-power resistor, as explained above will work without transients.