If your career depends on the gear, a hard case with cutouts. It's a pain in the ass, but it will keep your gear safe from small drops and bangs. As well it will keep it safe from water.
With planning you can have just the right spaces to make sure everything is quick to setup and break down.
Counter intuitively the more remote the location the more important a case is. The more bumps, bangs, thuds and moisture will happen remotely when hand carrying gear over a distance.
However cases are a physical pain to lug cases around.
Have you tried shooting a higher frame rate, or faster shutter speed? I am not sure if that would help, but it may.
to go a bit off topic. My two cents are that a dedicated video camera is worth having lower specifications.Much of what you need to kit out a DSLR will be built in.
My Canon experience is with the c300 Mk i and II. Using a video camera to shoot video is a much better operator experience. Having built in ND's, monitor, rally record,Canon log, XLR audio support , audio and video scopes makes shooting as a one man band easier. I spent a week in Hawaii shooting the c300(i) with EF zooms. I could not have done as well with a DSLR having to lug separate audio gear and all the ND's I would have needed.
I haven't shot with the 5Dmk3 or 4. Looking at specs along the MK4 is superior to the MK3 with the ability to record 4:2:2. They both lack log recording and that may be an issue post, but maybe not.
I haven't seem much Canon Dslr footage over here. I see more A7s(1 and 2) but still not that much. What I have seen of the A7Sii impressed me enough to pick one up.
The uncoated lenses have such a specific look and the lenses are so expensive it seems better to rent these than own a set like this.
Couldn't much of the look be achieved by adding a piece of uncoated glass to filter tray?
One of the big things you will want to considers, especially for commercial work is what codex and bit rates are required for your clients delivery.
Typically for video clips, online only stuff, you can shoot anything you want. For commercial work, there may be some requirements on native resolution, codec, bit rate and color sampling.
One of the reasons the Canon c300 took off like a rocket is that it met the minimum requirements for nearly universal broadcast delivery. (50Mb 422 1920x1080)
Call around to some of your potential clients, as well as distribution outlets and find out if they have requirements like this.
A camera like the c300 will meet most HD deliverables,and can be had for in your budget used. It will work with your existing Canon glass without issue. It also has professional audio inputs and can be rigged out professional for not a lot of money.
If you don't have that limitation, the A7sII is a great little camera and can be rigged out to shoot for around $4000. You will need to add budget for sound recording. The rode is good for scratch audio, but you can also add an on camera XLR adapter for $700. The Zoom f4 is an external recorder, but should work well. (I have an h6 and want a more video centered solution)
The Sony FS5 is also worth a look as a modern, video camera. though it may be out of budget when rigged up. However it supports lots of professional codecs.
Most cameras will be able to use ef-mount lenses with an adapter, so you can keep your existing glass collection with a metabones adapter. You can build a collection one lens at a time that way. A set of cheap primes will be thrown out and is a waste of time and money. Canon L glass is well regarded for still and video captures. I have the Canon cine primes, based on the L glass and think they are great.
People have a strong preference for primes, but a good zoom lens will handle more situations early on. You can always rent prime lenses when you get a job big enough to
warrant them. Or if you don't take my advice and pick up cheap primes like the Rokinon, pick them up over time for specific jobs and lengths.