If the vendor can't satisfy your requirements based on ASME cycles, then they can't satisfy your design, period!
You'll need to sign up for a free ID, but if you check out MDP16 here
http://www.pathwayb.com/index.html Go to LITERATURE on the side bar, TECHNICAL PAPERS, MATERIAL DATA PAPERS, and choose MDP16, you will find a write up on the difference between ASME and EJMA cycles.
Basically, ASME cycles are designed to insure that the design will survive the number of cycles specified. If you call for EJMA cycles, they are based on a fatigue curve fit that goes through the failure points, meaning some failure occur above the level, and some occur below. If you specify EJMA cycles, you need to specify a higher number to give a safety factor.
Essentially, if you need 26500 ASME cycles (which you do, if that is how many cycles your system must undergo), then you would need to specify somehing like 50000 EJMA cycles to get a design to do what you need.
Now that's that is out of the way, I think you're going to have a very hard time getting an expansion joint designed to handle that kind of cycle life. I'm guessing that your number is to be the cycles for a 20 year design life.
What you are probably going to have to do is discuss with your client what the planned interval is between shutdowns and see if you can get the joint designed for that cycle life, with plans to replace the joint as part of the planned shutdown work.