Not a bad idea, Jannik, and many have done it before you. Basically you are paying the same money, get an FAA and JAR raing at the same time and 500 hours on type.
Do you have an FAA or TCA CPL/IR?
Many JAR authorities will let you put an ICAO rating onto your JAR licence (of course limiting you fly a certain register).
Remember to budget for a JAR LPC once you have the 500 hours, which may not be cheap. However, many JAR OPS airlines will accept the FAA rating together with your JAR licence and 500 on type, as their conversion course will cover the LPC anyway.
Speak to Eaglejet directly, they help with ratings. They sent many people to a company I used to work for (non-JAR country), needless to say, that they all have paid jobs now.
many 737 TRTOs around, they should be well below USD10000.
try google for these and more:
here,
here
and here.
hope this helps, IP