First, I assume you are on a well? Please confirm.

In your situation, a flow rate is going to be difficult to obtain, but we can estimate based on the fact that you have a whirlpool. Is this a multi-person hot tub or a smaller single-person jet tub?

Have you checked to see what your neighbors are doing for water treatment?

What did the independent water treatment company suggest?

What did Culligan suggest?

If you have access to get a water sample, run water into a clear glass container:
Is the water clear when it initially comes out?
If initially clear, does the water develop color if left to sit?
Are there visible particles in the water?
Does it have an odor?

It is just plain silliness to leave the cold water to your kitchen sink unsoftened. Your sink will be a mess with hard water deposits and very difficult to clean. Your inclination to install an RO is MUCH BETTER. Plumb RO water to the ice maker and have a separate RO faucet at the sink. RO water for drinking and cooking. Softened water for washing.

Well water, no matter how clear it looks at any given time, has a tendency to have periodic particulate matter. Particulates are bad for softeners, so filtration should be installed upstream from the softener. A pair of "big blue" type particulate filters, plumbed in parallel and valved so that you can run through one while the other is changed is highly recommended. Place a pressure gauge at each end so you know when to change the filters. If you can find them, I prefer a clear housing so you have a visual indication as well.

Yes, we are on a well and it's a one person whirlpool. The water is very cloudy but we were told that may be because it's new. Not sure. No order and yes we do see particles after the water sits a bit. We did not notice and color develop. Our plumber is going to shock the well and after it sits a bit we will flush and have it retested.
Thanks for the info on cold water being softened and I will check out the big blue filters.