You may have a quick and dirty way out. It seems that VN offers online check in up to 24 hours before flight time. Considering you're starting out from Darwin, you ought to be able to do this and have printed boarding pass in hand before setting out. I would strongly suggest you call VN and see if there is any reason why you would not be able to do online check in. Alternatively, it appears that the ground handling agent at Changi for both MI and VN is SATS. This is a hail Mary because normally the transit desks only handle airlines for their specific terminal but maybe the MI agent at the T2 transit desk might be able to help out by issuing your boarding pass as an exception. The number for SATS in Singapore is +65 6542 5555.
This next part is a comment on your prior Changi transit. Further the point the transit desks only handle airlines for their specific terminal, this was not an issue in the past. When there was no T4 this was no problem as going between terminals on airside via Skytrain was easy. For you it was even easier since you were going from 3K to TG; both airlines operate from T1.
In case worse comes to worse, you can enter Singapore legally and check in at T4 from ground side. Note that check in closes 50 minutes before flight time. Knowing Changi, this timing is tight but potentially doable assuming your flight arrives on time and you do some running.
Airline, Check-in Counters - Singapore Changi Airport for the confirmation of SATS as ground handling agent and VN last check in at T4 from ground side.