Main matching via Graide Color Match and fine tuned with Graide Color Curves.

Here is another one with VP core tools. First VEGAS Levels FX to adjust black, white, gamma. Second VEGAS Color Match FX. Third VEGAS HSL FX for fine tuning hue.

Away from trying to match your A7S reference I prefer this one as to me it looks more natural. It's done via Graide Color Curves only.

I would say this is twisting 2 "cold" things in one hotpot: Colorspace converting and color grading. 

Such source footage is challenge for none; it's neither fish nor fowl. 
It usually invents an "advantage" from the Rec709/sRGB space: The LOG gamma, in times that our hardware still suffered their limit. 

Today, our camcorder can capture not only in real wide color gamut also even with HDRange. 
But this aside. 
Let's roughly say, LOG-footage belongs to WCG, too. 

So, as you can see, such high quality source footage must be converted in the post editing for their final output: SDR or HDR. 

As for SDR production, the LOG-gamma could take some advantage compared to true StudioRGB material, e.g.: 

But as for "normal" post job, you must cut this advantage away, e.g. here you lost cloud details in highlight: 

And as for HDRange production this LOG-gamma can reach about 500-600nits and remain all highlight content. 
Sure, it's not true HDRange footage. 

Refference:

Grade (by eye-ballin):

Project settings: ACES cct, sRGB transform

Media properties settings (color space): GoPro - ProTune Flat - Protune Native

Wheels 1:

Wheels 2:

SDR grading, emphasize green:

All work done in the Color Grading panel

First I sampled the blacks with the VP19 Specify a Black Point tool

Next, I sampled a cloud with the Specify a White Point tool

Then, I bumped up the Green Color Curve just a touch on the low end

Finally, I set the Input Level to .96

Same here as @gary-rebholz. except for Step #4, where I emulated a Photoshop Gray Point Picker (a feature already suggested) by substituting the Vegas WB picker.

Last, an overall Gamma adjustment was made. My goal is preserving some detail in the clouds, which is difficult without a mask. [EDIT /] In the course of testing this, I cam across the Highlight/Shadow preset in the CG Curves section. Turned out to be just what the Dr. ordered.

Maybe @Grazie will have time to show us what is possible with a Highlight Mask in Colorfast2.

I'd take your Leeming LUT an add an S-curve to it to bring down the shadows and create more tonal separation. Then you can adjust any colors selectively to taste (skies, green, etc. if you have a particular preference). It is flat by default to give you a good starting point for further work in post.

I'd join the fun here, but am away and don't have a proper calibrated screen. For actual work I'd use Graide Color Curves (as Marco did) as it's the most time-efficient. On my laptop monitor the A7S looks bluish, which may be accurate and how Sonys often look, but not how I would want the final image. I would warm it up slightly.

I probably wouldn't use the black or white color pickers as it clips too much data on both ends. Better to send end points conservatively using levels or curves (or input/output on the color grading panel) and then increase contrast with curves instead of clipping anything.

Here's Marco's 1st edit with a little more contrast added and a little green removed via Color Balance fx:

I think in terms of matching the Sony reference, this is pretty good. The sky, clouds, and green grass are all very close, and the edit doesn't look as destructive as the Leeming LUT. Interestingly, the weeds in the foreground are completely different. But for overall matching, the Graide plugins are giving a credible result.

Adis' eyeballing is very impressive:

Overall accuracy matching the reference looks good -- especially impressive is the foreground vegetation, which looks almost identical. There's a red roof in the background which matches. Maybe there's some kind of shift happening in the sky, and a slightly muddy look, but the GoPro is never going to look as pretty as the A7S.

I had no idea you could go into ACES and flag source clips as Protune. Thank you so much for pointing this out! Another tool in the toolbox.

Musicvid's sky, clouds, and water all look natural to me, and a very bold and dramatic look overall:

"That's what sky is supposed to look like," my eye says. I think the actual look of the scene was quite a bit closer to the Sony's capture, but without that reference, at least the top half of this is a very pleasing take.

The vegetation and the darkness in the grass does change the character of the look -- has the feel of a cold, northish place, rather than the sunny SoCal scene it actually is. If I brighten this, I think it has one of the most credibly natural looks about it (the colors "seem" accurate, even if they aren't) -- except for the foreground vegetation, which has gone magenta.

Would be interesting to see how Musicvid would have rendered this if he had been onsite and seen the scene in person. Maybe he is used to chilly beaches? :)

Gary's edit gives us really nice greens on the golf course:

The midtone contrast here is good, so there's a crispness and a vividness to the grass that I find very appealing. This might be my favorite in terms of the green grass, but the other colors aren't holding up as well.

I'm going to add one more of my own here:

This is Vegas' White Balance plugin again, but this time I'm using the temperature slider to back-and-forth until the color looks neutral, then the tint slider (back-and-forth, again) to neutralize the grays. Then adding Leeming's professional LUT (which converts Protune to REC709), some contrast and saturation.

Of my attempts, I think this is the best color balance overall...but look at how degraded the image is. This is the problem I have with Leeming's LUT -- it's so destructive, particularly in terms of color noise. I would love to be able to get to this point natively, and hopefully not create so much color noise.

Musicvid's sky, clouds, and water all look natural to me, and a very bold and dramatic look overall:

Thank you for the kind thought. When looking at it, my question was, "What are we interested in looking at?" My eyes drew me to the horizon, but looking at it again, maybe the fairway is a better choice.

"That's what sky is supposed to look like," my eye says. I think the actual look of the scene was quite a bit closer to the Sony's capture, but without that reference, at least the top half of this is a very pleasing take.

That's one advantage to using a neutral picker tool.

The vegetation and the darkness in the grass does change the character of the look -- has the feel of a cold, northish place, rather than the sunny SoCal scene it actually is. If I brighten this, I think it has one of the most credibly natural looks about it (the colors "seem" accurate, even if they aren't) -- except for the foreground vegetation, which has gone magenta.

I, like a few others here, just didn't take kindly to the greenish haze; so I chose my neutral pick point nearer the beach, knowing the foreground would go cooler.

Would be interesting to see how Musicvid would have rendered this if he had been onsite and seen the scene in person. Maybe he is used to chilly beaches? :)

And that of course, it the gist of grading Protune.

"Memory" colors aren't everything; they're the only thing.

Try applying the LUT using tetrahedral, and test 32 bit video mode which reduces banding in my experience.

Shooting flat on small sensor action cams has its limits. There is no 10 bit option I assume?