I found a pretty reliable and easy way to kill the Alpha Bush Devil in Dinkum.
The main goal is to build a dirt wall two blocks high and three wide near the spawn location of the Bush Devil. You will have to build the wall while dodging the Bush Devil so remember to take your time and be patient as the attacks do a lot of damage.
After constructing the wall, you can jump on it and the Bush Devil should immediately deaggro and go back to sleep. If it doesn’t, add one wall segment perpendicular to the wall. If the Bush Devil is too far away, build more dirt wall segments until you are in melee range. Keep in mind the Bush Devil can climb wall segments one block high so you may have to run away and come back if it manages to climb up.
After this is done, you can simply melee the beast until it dies. It will be stuck in a permanent loop where it wakes up and then immediately goes back to sleep.
Hope that helps!