Value for $ 3 or 5 axis gimbal for DSLRs?

Your weight requirement is very heavy for a single handle gimbal at 1600g. This CAME gimbal maxes out at 1200g. http://www.came-tv.com …-with-encoders-p-673.html Also using a zoom lens with shifting weight will be bad for stability.

is there no other single pistol grip gimbal built for DSLRs?

Even if I were to pair it with a 16-35mm f4L IS, the total weight would still be 1295g - exceeding the max payload.

Really quite a pity - because if I 'downgrade' to a GH4, the low-light performance would be too poor to shoot in dim conditions without additional lighting.

I'm planning to use it to shoot my vacation videos - hence the lighter the better (without compromising too much performance).

Also, I thought that the brushless gimbal is supposed to self-balance regardless of how the weight shifts (am i wrong). Otherwise, what is the benefit of using this compared to traditional steadicams?

I think you missed the point of a "vacation".

Good low light performance, gimbal stabilization, small, and lightweight just don't all work together.

Your 24-70 zoom is nice, but half the weight. You could still use the CAME single handle gimbal with the canon 50/1.8 and 28/1.8 lenses.

This SteadXP technology will also mature soon for SW gimbal style stabilization in post.
https://fstoppers.com …gopro-video-footage-86142

It also might be easier just to use a different approach such as:
https://www.cinema5d.c​om/hands-dji-osmo/

You don't really downgrade by going to a GH3, ISO performance is not everything when it comes to video.

Heck some cameras that pros use are limited to 800.

Check out the nebula 4200. You can also get a nebula 4000 used for pretty cheap.

Nebula 4000