Here's the thing:
You can make it so that the second tier displays Royal Archers (by setting a different sequence in Ugrades), but then you won't be able to upgrade Archers to Royal Archers, or Royal Archers to Archers.
Or you can do it the way I did, and then you can upgrade Archers to Royal Archers, and Royal Archers to Archers and Snipers.
What's the problem with doing it like this?
For the white-gold upgrade, specify Royal Archer first in the upgrades, and then the red upgrade with the Archer.
For the red upgrade with the Archer, specify the white-gold Archer first, and then the red Archer with the bow.
For the red Archer with the bow, specify the basic creature of the red Archer. And for the Royal Archer, specify the white-gold Archer.
It seems like it should work perfectly, right?
Added 8 minutes later
If, for some reason, this doesn't work, you can do it in a similar way to the example with the Gremlins.
That is, for the third level of upgrades, specify a basic, completely useless creature. This creature will not appear in the recruitment in the castle, and will not appear anywhere at all, but it will have the correct upgrade order for the third level. For this, you will have to sacrifice a creature from the non-castle faction, but it's not a big loss. You probably won't need all 999 of them.