I saw a CalMan video on calibrating an LG monitor and was wondering how I might use ColourSpace to get similar results.
The demo showed how CalMan was interacting with the monitor to do multiple iterations of profiling with relatively small patch sets while adjusting the LUT after each iteration. Without having a monitor that can use LUTS (or a LUT box that interacts directly with ColourSpace), is there a way to apply a LUT, re-profile, then combine the LUTS for a better result?

That sounds like the 'Autocal' iterative guesstimation Calman uses.
Such adjustments are nothing more that automating, via guesswork, the available manual adjustments of any monitor.
It is not adjusting a 3D LUT, as that is impossible.
(To attempt that would cause major image artefacts & banding, etc.)

If a monitor cannot use a 3D LUT, performing a user manual calibration will nearly always provide better results than an 'autocal' approach.

If you can use LUTs, then ColourSpace has many different ways to ensure the LUT is the best possible for colour accuracy.
You can use Augment, or a User Defined patch set that focusses on different colour locations.
Or you can swap LUT components, such as the Grey Scale, etc.
Or concatenate LUTs.

All this is defined in the User Guides.
For example, ColourSpace Advanced Operation.

Steve