1) NO ONE should own a Crossfire and NOT have a spare RCM. They plug in without tools, they fail a LOT and they are hard to find locally at times.
So, Order one today, if it fixes it, fine, order another to keep as a spare. If not, you should have one anyway. YOU DO NOT WANT a "Crossfire" RCM, you want an RCM for a 2003 Mercedes SLK320.
The supplier's website will tell you that it won't fit a Crossfire ignore that statement.

2) Sorry, but your text is a bit confusing to read. Let's just look at this:
Crossfire will start and then immediately die. When I try starting it again it will not start and it just keeps turning over. I’m getting no fault codes. The car has a new battery.

What is the fuel pressure? Clearly, the RCM's ECR output is working, otherwise, the starter would not engage. But is the RCM's output to the fuel pump working? Is the fuel pump working?

I"d buy a cheap fuel pressure test kit and see what the fuel pressure is at the end of the rail, 58psi +/-4psi is expected. If you can not hear the fuel pump run for about 5 seconds when you turn the key on, the suspects are:
Bad Fuel pump relay on the RCM
Bad fuel pump
Bad filter/regulator
Bad fuel pickup in the tank

The EASIEST thing to check is the fuel pressure, it's like checking tire pressure, cept if you allow a leak you can set the car on fire - but do not be intimidated, just do not allow smoking within 50 feet of you at the time.
I have one of these - cheap but they do the job: