Actually, I'm not sure if I want Alolan Dugtrio to get Arena trap, for the simple fact that if it does get it, then nobody will bother with Kanto Dugtrio ever again.
The Alolan forms are an alternate way to play some of the original Kanto Pokemon yes, but they shouldn't become a complete replacement for them either.
Alolan Raichu gaining the psychic type means that it gets resistance to fighting and psychic attacks (along with a secondary STAB to take out fighters and poisons), but it also opens up major weaknesses to Shadow Sneak, Sucker Punch, and Knock off. Kanto Raichu may not have the extra STAB and resistances to play with, but Kanto Raichu still only has one weakness, and an immunity to electric instead of a resistance. Both variants of Raichu provide something different depending on what mood you are in, and you can't go wrong with either one. Its just a matter of preference.
Ice typing helps out Sandshrew and Vulpix lines, but the original forms are still just as viable. Alolan Ratatta loses Guts as an ability, which still gives Kanto Ratatta line a role as a fast normal type nuke.
But if you give Alolan Dugtrio Arena trap, and a secondary fire typing, and a couple of new moves, it would definitely be awesome, but then Kanto Dugtrio will collect even more dust.
Personally what I would do is keep Arena trap with Kanto Dugtrio, as that's the one thing that it can do that nobody else can (Mega Gengar and Gothitelle really hate the ghost and dark priority moves, and can't block any of the offensive switch out moves). For Alolan Dugtrio, I would pull a "Scizor" and rearrange the base stats of Kanto Dugtrio.
Kanto Dugtrio's base stats are 35/80/50/50/70/120. For Alolan Dugtrio, I would take 20 points out of special defense and redistribute them to attack, giving Alolan Dugtrio a spread of 35/100/50/50/50/120. Being half fire type (in theory of course) will give Alolan Dugtrio the ability to use flame charge, and boost that already high speed even higher, outspeeding nearly the entire game. Throw in a fire punch, and Alolan Dugtrio is pretty much set (Ground/Fire typing is only weak to water and ground compared to Ground which fears water, grass, and ice).
The ability that I would give to Alolan Dugtrio though is Flash fire. That would not only help boost Alolan Dugtrio's damage potential, but also add an immunity that Excadrill wishes that it had.