As a general advice I'd say that you don't have to feel obligated to stick too closely to the lore, especially where it is a bit thin. Even real-world militaries don't always map well into Aurora, so certainly game lore you can expect to fit in even less without some wiggling. Alternatively of course you can play with the rule of making cool lore stuff first, and making it look neat in Aurora second. Based on what you're doing here I'd suggest looking at the initial post of Steve's
Imperium of Man fiction, because he uses some fairly simplistic ground formations based on WH40K lore (which is far from realistic to say the least!) that are not exactly optimal but get the job done reasonably enough. It's a good example of how to blend lore and Aurora mechanics if you just want to shoot things and don't want to get super technical about it.
Infantry Battalion: you might try treating the infantry/marine battalion (company? At 1,000 tons and ~140 soldiers this is a company-sized formation in most militaries) as a mechanized infantry unit, which means you have the foot riflemen transported and supported by armored fighting vehicles. In that case you have Marines (INF-PW), Warthogs (LVH-CAP), and I believe there are close air support vehicles in the Halo games (Banshee?) which you could use as LVH-MAV for an anti-armor capability. "Vehicle" can mean whatever you want!
As a general pro-tip, consider having 2x command elements in a formation instead of 1x, so that if one gets killed you can still receive bonuses from higher HQs (although this may be bugged right now).
Armored Battalion: glancing at (Googling) the lore, the Scorpion would be better as MAV/(H)CAP as it has a secondary MG rather than a cannon. Reasonable change and VEH-MAV/CAP is a pretty standard early-game main battle tank in Aurora so it works well.
Putting LOG into a formation like this can work two ways. On one hand, it can give the formation extra staying power particularly on the offensive; on the other hand, LOG is not a combat element and doesn't contribute to killing things directly so could be a hindrance that gets shot up and dies quickly. For a formation this small I probably wouldn't have LOG, but for a 5,000 ton battalion it would be okay to have some. In either case you want a lot of LOG in the rear echelon superior formation to be your main supply source.
HQ Assets: I'd generally avoid mixing HB and MBL and settle on one or the other. The MBL on the Mammoth could probably be some size of AA weapon instead (also matches the lore?). Otherwise, without digging too much into the lore I'd just say that you really want either a bigger formation (5,000 tons or more), or you want to split out the LOG into a separate unit - 1,000 tons of LOG gives you 16 Sandcats and a total of 8,000 GSP which will support a pretty good amount of troops for a good, long offensive. In either case, that will let you figure out how you want to fit your HB/AA/HQ together separately if you stick with 1,000-ton formations. Note that SHV/UHV are
extremely heavy, so I'd probably not recommend using them in anything smaller than a 5,000-ton unit just for practical reasons. Generally I'd only use SHV in a front-line breakthrough formation, as the massive weight and armor isn't worth a lot in the rear echelon, but here we have lore to contend with so we will use it and be badass about it.
So keeping it simple and sticking with 1,000-ton formations, one idea would be something like:
Infantry Battalion: treat it as a mechanized formation and the vehicles fit in perfectly
100x Marines (INF-PW)
8x Warthog (LVH-CAP)
6x Banshee (LVH-MAV)
2x Mongoose (LVH-HQ1)
Armored Battalion: The only change I'd make here is MAV/CAP on the Scorpion and move the LOG to a rear formation
10x Scorpion (VEH-MAV/CAP)
5x Cobra (VEH-LAC/LAC)
2x Mongoose (LVH-HQ1)
HQ Assets: Separate the LOG and you have more room for your other things. It's not the neatest-looking loadout, but HQ formations are always ugly and this will get the job done just fine.
1x Mammoth (SHV-MAA/HB/HQ6) - HQ6 to support the above plus a logistics formation
1x Elephant (SHV-CON/LAA/CAP) - change from LAC to CAP since this is again a MG I think?
3x Kodiak (HVH-HB/HB)
Logistics Support Battalion: 16x Sandcat (LVH-LOG), there's not really a need for a commander here so no Mongoose.
If you decide to scale things up to 5,000 tons per battalion, you can mostly just multiply by 5x and be fine (keep only1x or 2x HQ as the HQ element size will increase). Again, this is just one idea for how to get reasonable formations from the units you're working with, by all means don't just do what I've done if you have a better/more fun idea.