I use Gates Lock-On / Push Lock or the Dayco equivalent that is rated for 300 PSI. It can be used without clamps if the proper fittings are used, these fittings will have a yellow plastic hose protector ring in the middle of the fitting. 5/16" ( 8 mm ) and 3/8" ( just under 10 mm ) are common automotive trans cooler line sizes.

For your fittings, you will need to use clamps. How are you attaching to the car side of the lines? Specifically where did the hose fail? If you are sliding the hose over a flare fitting, the sharp edge can cut the hose. Cutting the tubing to a raw edge will cause a similar failure. I use the first step of a _Double_ flare or a partial _ISO_Bubble_ flare tool to form a barb so the hose / clamp has something to grip to.

In situations where I can't make a barb, I slide the hose on 3" and use 2 evenly spaced clamps at normal tension. While we are on the subject of tension, if the hose is extruding through the clamp slots, it is way too tight. For most 5/16 and 3/8" ID hose, a narrow band SAE number 4 clamp is sized properly.

Stainless braided is only used for severe abrasion resistance and many times is too stiff for low pressure applications causing fitting / cooler failure. If you need abrasion resistance, get steel spring covering ( like factory ) , plastic spiral wrap or nylon braided covering. All of these are usually available at a hyd hose store for almost no $.

As a side note, where in the fluid circuit are you plumbing the cooler into?

If trans outlet / external cooler then rad cooler, the fluid won't be cooler than engine temp. This is good in winter but less good in summer as it leads to high trans temps.

If trans outlet / rad cooler / external cooler , the fluid will be cooled below engine temp. This is good in summer but not so good in winter as the trans won't boil off moisture.