If I am understanding correctly (a big 'if'!), the role of a regiment is primarily administrative/mobilization related, so it shouldn't show up in an order of battle but you can still roleplay it effectively with some creativity.
A combat division order of battle would look something like:
XX Division
X Infantry Brigade
II Infantry Battalion
II Infantry Battalion
II Infantry Battalion
X Infantry Brigade
II Infantry Battalion
II Infantry Battalion
II Infantry Battalion
X Infantry Brigade
II Infantry Battalion
II Infantry Battalion
II Infantry Battalion
Other formations...
For the sake of simplicity I'm not looking at the artillery, etc.
And each battalion would in turn have come from a regiment, which is an administrative division not present in an order of battle. In a peacetime situation, with the division deployed somewhere (India, Egypt, Gliese 487, etc.), the battalions would basically be rotated with others from the same regiment so each could receive leave time, training, manpower rotation/turnover, and so on.
So in Aurora terms if you have the 1st Infantry Brigade stationed on Luna, made up of the 1st Battalion, Martian Regiment and 2nd and 5th Battalions, Terran Regiment, then in six or nine or twelve months or however long a transport flotilla would arrive bearing the 2nd Battalion, Martin Regiment and 3rd and 8th Battalions, Terran Regiment, which would replace the former three battalions in the 1st Infantry Brigade OOB. The former battalions are then shipped back to Mars and Earth where they would (in RP terms) undergo training, receive leave time, see some personnel turnover, maybe receive equipment upgrades, and so on.
If my understanding is correct...that sounds like a lot of extra work to do manually just for RP goodness. The simple solution is probably just to use Battalion names to represent regiments and leave the numbering off, so instead of "1st Battalion, Martian Regiment" you could name it something suitable - "Martian Battalion" to be concise, or "Infantry Battalion, Martian Regiment" to be wordy. A bit awkward but it gets the point across and you can imagine the rotation in your head.
A different solution would probably be to have the regiments as large HQ formations with a bunch of battalions under them, and then for wartime pull those battalions out of the regimental OOB and organize them into brigades and divisions on an ad-hoc basis. That could be interesting later on as you put GU training facilities on different bodies and have Terran, Lunar, Martian, etc. regiments on each one instead of the traditional garrison formations.