Considering getting the LS300 - I will of course get the metabones so I can use my existing lenses.

I'm not entirely clear about the re-mappable sensor. I think I can still use my 24-70 Canon lens, but at 1.5x crop it'll be a 36-105 equivalent.

That's not wide enough for run and gun use.

Any recommendations on a (hopefully native) walkaround zoom lens?

If you get the Metabones Speedbooster you will end up with a similar FoV to what you are use to (assuming you're shooting 5D/full frame)

A native lens would be something like the Panasonic 12-35mm - the sensor will have to be set to M4/3 size which would give you the same FoV as the 24-70mm on a 5D.

So if I get the metabones, and I change the sensor size, whatever lens I'm using will be true to it's origin? I thought the crop was a minimum of 1.5x - couldn't get less than that (because of the MFT sensor). No?

The LS300 has a Super 35 sized sensor (about the same size a 7D) which can be made smaller with its variable scan - but it has a M4/3 lens mount (this will cause a lot of confusion)

A Speedbooster would shrink a full frame lens image circle down to about this Super 35 size so your FoV would be about the same as on a full frame camera (with an extra stop of light).
Metabones also make a straight EOS to M4/3 adapter which has no optical element so won't change the image size, meaning that it will produced a cropped image compared to a full frame camera.