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The core of my list is a Battalion of GK, a Battalion of AM, and a Catachan Shadowsword with all the flamers.

GMDK, psilencer, psycannon, hammer, teleport - 290
GMDK, psilencer, psycannon, hammer, teleport - 290
GKSS, 5x, falchions, psilencer - 107
GKSS, 5x, falchions, psilencer - 107
GKSS, 5x, falchions, psilencer - 107

Company Commander, Warlord, Aquila - 30
Primaris Psyker (Nightshroud and Barrier) - 46
Infantry Squad, 9x, HWT mortar, flamer - 53
Infantry Squad, 9x, HWT mortar, flamer - 53
Infantry Squad, 9x, HWT mortar, flamer - 53

Shadowsword, 4lascannons, 4 twin heavy flamers - 604


So the question is, should I take a third GMDK, or should I take some more supporting options, namely Greyfax and a unit of Elysian plasma+Elysian Commander?
If I do that, I'd drop the Primaris Psyker and take two Astropaths instead, allowing me to get both buffs onto the TANK instead of just one a turn. Greyfax is also a good combo with Purge Soul and can deny overwatch when my GK wants to charge a scary flame unit.

Based on the numbers you have given you don't have enough room for a 3rd GMDK even if you drop your hammers to swords/fists.

You may just want to drop the psilencers back to normal stormbolters so that the entire unit can benefit from the psybolt stratagem.

You may also want to look at adding a unit of 5 Interceptors with falchions. Take them with vortex of doom and use the 3d6 take the best stratagem on clumped up enemies (cost 125 points).

Is there any reason for needing Elysians as opposed to any other AM regiment?

I'd be at 2030 if I added another GMDK onto that list. I'd drop the flamers from the guard groups and do some more wiggling to get me that 30pts. So it's very doable.

Using Psybolts on a 5-man unit isn't the best investment of CP imo.

One of the GKSS has Vortex, and is a suicide squad that drops by the best target for Vortex. And yes, I always use Channeling and save my reroll that phase to try for the 12+.
At this point in time, I'm unsure how I could actually fit a unit that costs 125pts into my list. I assume this would be in lieu of the Elysian Plasma?
But if I'm dropping the plasma anyhow, why not just take a 10-man GKSS? Then I legitimately COULD use Psybolts and feel like it wasn't a waste....
But I already have so much anti-infantry, is it really necessary to take MORE stormbolters instead of some plasma?

Taking Elysians instead of Tempestus is 8pts cheaper for the exact same choices, for some reason. And if I'm already breaking a pure detachment (by tossing Greyfax in there) I may as well break it further to save a few points.

I agree that it isn't that cost effective to use the stratagem on a 5 man squad but thought that I would suggest it anyway just in case you weren't aware of the stratagem.

If you're looking for anti-vehicle you may want to look at a Stormtalon with twin assault cannons and twin lascannons. It's relatively cheap at 204. You could also consider some anit-flyer dakka with a Stormhawk with 2x Assault cannons, Icarus Stormcannon and skyhammer missiles for 175. There are a lot of units with the "fly" keyword and the Hawk will tear them up pretty well. Both flyers can take advantage of GK abilities (like Astral Aim or reroll auras).

You may also want to consider Draigo with his aura and abilities he makes a pretty good anti-vehicle weapon as well.

I'm a new GK player so please take my advice for what it's worth.

Option 1: 3x GMDK


Option 2: Plasma