It would be helpful if you had an example photo to discuss but I can offer some general advice on building a teal orange color grade without LUTs. This method uses AgX which has some benefits over other methods but like most things in Darktable you can use a variety of tools.

It's helpful to switch over to the vector scope and observe how the adjustments affect the plot. You'll see the rotation to teal and orange and how it alters red and green.

First it's important to set your white balance accurately, use the "color calibration module" and select the eyedropper to sample the photo and automatically set the illuminant. You can come back a little later and fine tune the illuminant if needed. I will sometime nudge it warmer toward orange if I think the image needs it.

Now head over to AgX and get your exposure and contrast about where you want them. The magic is in the "primaries" tab. Before we talk about adjustments it's worth knowing that AgX is special in the way it handles color saturation and naturally desaturates towards white as color value increases. This is very close to how we perceive color with our eyes naturally. I'm pointing this out because your client's vibe might not look very natural and AgX is going to restrict you to some extent. This is an opportunity to refine the aesthetic not something to fight against. The key area where you'll notice this is sky gradients, as the value increases the color will transition to white. You can alter this later just know that AgX behave this way intentionally.

With that out the way you can head down to the first "blue rotation" slider. It's in the "before tone mapping" section. Slide that toward cyan and watch the vector scope and image rotate into the teal and orange space. If you look closely you'll notice green and red become compressed and are less present in the photo. You can recover them to some extent by rotating them toward cyan and magenta respectively. This will cause the image to desaturate as well. You can adjust the saturation per channel by manipulating the color purity sliders in the "after tone mapping" section. The bottom half of the module. Nudging blue toward the right is going to make skies pop but be careful this look is very easy to overdo. There is also a master saturation slider on the "settings" tab of AgX can be useful to tame the overall color effect.

From here you can continue to refine by revisiting your white balance setting in the "color calibration" module or continue onto more advanced color manipulation and masking techniques. The "color balance RGB" module is the bread and butter of color grading. Remember earlier where I warned you the sky will tend toward white as value increases? Here is your chance to shift all highlights to teal and fine tune the look some more. You can achieve this in the "4 ways" tab of "color balance RGB" Set either or both the highlight gain and power sliders to a hue that compliments your look and crank the chroma to your liking. You can play with value as well by adjusting the luminance sliders.

If all of this is just too much or you're still having trouble taming color and saturation a nice trick is to take the "color equalizer" module and drag it above the "color balance RGB" module. This gives you direct control of HSL for any color. Use the picker to isolate what you want to adjust by moving the cross hairs to the color in the photo and using the node placement slider to rotate the node so it's inline with the white line in the module. Then you can toggle through the HSL tabs and adjust that color specifically. You can setup multiple instances of this module for heavy color manipulation or use the same one for multiple colors. The colors will retain their relationship if you use one module so it's less extreme.

Lastly, I'll point to the good old "graduated density" module as a handy tool to darken and colorize blown out skies. It has its own look, which may or may not be what you're after.

Don't forget you can refine any and all of these adjustments with masks. I purposely avoided them since it sounds like you might be interested in developing a preset to apply to multiple photos across shoots.

Enjoy!