Based upon the information you have provided I see no indication that the rings are bad.
Bad rings will almost always give you those two symptoms, if they aren't sealing well against the cylinder walls you burn oil. That's it.
Another point of interest is that you have a shim and bucket valve train. More on that later.
Here is an image of a shim and bucket valve train. Notice that there are no rocker arms or push rods. The cam acts directly on the bucket and shim which acts directly on the valve stem. You adjust the valves by changing out different shims to get the appropriate valve clearence. In this image, the shim is below the bucket. This is what the CBR has. It's called an Under the Bucket Shim.
Other posters have told you to buy new pistons and rings.
If you buy new rings, you will need to have a new nikasil coating put on all of your cylinders. No shops that I have ever encountered offer this as a service. You have to have them either ship it out their provider or you find one and send it off to them.
If you bore your cylinders and put the 1st step overbore into the bike you may as well send it out to a place that does the nikasil. If you skip the nikasil, the rings will quickly wear out and you will have an issue.
Here is a link explaining the virtues of nikasil.
BUT, since none of this really applies, let's move on.
Even though your compression went up with the oil in the cylinder that is not proof that the rings are bad. It just seals them up a bit better from normal compression loss and helps to increase the compression.
The issue you are experiencing is out of adjustment valves. The symptoms are these depending on the severity.
Poor low RPM response
What you are looking for is counter-intuitive. You are looking for no valve clearance as opposed to too much.
A shim and bucket setup will lead to zero clearance situations where the valve can actually hang open and there is a tiny gap between the valve seat and valve face in the head. When this condition arises the vehicle is hard starting, poor idling and general runs like a bag of hammers. One valve hanging open can lead to a very poor running/idling condition that improves with higher RPM. Running a compression test can give you breadcrumbs and hints but a leak down test will truly determine if this condition is occurring. It's a very common issue and is considered general maintenance to adjust your valves.
This happens as the valves get super hot and get beat in and then the engine cools, the valve shortens. So you get into a situation where the condition can get worse when the engine heats up and the valve doesn't make contact with face. It will actually hang open just a bit.
Along with the mechanical beating of the valve face against the valve seat, you get a gap.
All you need to do is adjust the valves. You remove the cams after measuring everything out with a feeler gauge and documenting it and then subtract the value you need to get valve clearance and goto the Honda shop and order up some shims. Be sure you put the shims in a cupcake backing pan or ice tray, something where you can keep track of where they go, exhaust cylinder 3 etc...