[JamOfBoy]

Me (70 cel)
Engineer
Railgan

[...]

Next turn the order is:
Railgan
Me

[...]

Is this classified as a bug or not? I'm legitimately asking cause I don't know.

I don't know entirely but I can hazard a guess:

When the summoner is alive, summoned units do not count for the turn order, taking their turns after all PCs/main participants. Youkai and robots will always go last.
UNLESS the summoner/engineer is dead, in which case they assume regular turn order rules.

You killed the Engineer on your turn. The turn order at the start of that round was calculated with the bot being disregarded in the turn order.
The game reads that as you going last for the round robin system, since the bot doesn't count for that round and the other participant died and didn't get a turn. Next round, it reads that you went last, and now that the engi bot is actively considered in the turn order, this means it goes first.