Any variety will develop the diseases if the conditions are right. They are only resistant, not immune to them. So it would be more productive to explore the options of preventing the disease in your particular garden.

They all thrive in cool, damp, shaded, and poorly ventilated areas so things like avoiding over-head watering, increasing sun exposure substantially (try containers that can be moved for example), thinning surrounding vegetation to increase air circulation and sun exposure, covering plants during periods of heavy rain, trellising, close monitoring and early detection, succession plantings, etc. can all help.

That said, Cool Breeze is touted as being very resistant. It's quality as a pickle is debatable but acceptable. Eureka is another and makes better pickles IMO.

But I also have great results with good old National Pickling. The disease issue may develop depending on my weather but with staggered plantings of multiple hills which you have to do anyway to have enough to make a batch of pickles at one time, I still have cukes coming out my ears.

Dave

I had a problem with my cucumber last season. I had about 24 plants which were very healthy but by the time the plant has flower and little cuke somebody kills the whole plant. I found the stem at the soil level is damaged but could not find who did it. I lost 10 plants and was frusterated and mad. By pure luck I had insectcide called stracecide i mixed two table spoons to a gallon of water made a circule around each plant and filled the circle with the mex. I did it every other day for a week. It was a desperate move but I am happy I saved 14 plants. I guess the chemical killed whoever was doing that.

County Fair is a great pickling cuke that is highly resistant to the bacterial wilt carried by the cuke beetle. Little leaf is also quite resistant and is also downy mildew resistant.

I've grown white spine variety called Jackson Classic (F1) with good success has resistance to Anthracnose, Angular Leaf Spot, Cucumber Mosaic Virus, Downy Mildew, Powdery Mildew, & Scab

Arkansas Little Leaf is a species cross with very high resistance to bacterial wilt and powdery mildew. It's open pollinated so you can save the seeds provided it is isolated from other varieties. Except for a few Asian cukes for the table, it's the only cucumber we grow because it's so healthy and dependable, and the pickles are great. After it has produced a bodacious crop it gets spider mites, but don't they all?

Here is a link that might be useful: Little Leaf at Southern Exposure