CE and FCC compliance
CE also Comes in different classes (A-E, I think)


No it doesn't but, some of the sub-testing/approvals (as specified by applicable EU directives) such as the EMC directive, have certain sub-classes i.e. part A and B emissions etc..

Blow is what this site says about FCC part 15 (Electromagnetic interference): -


  What about the new FCC Part 15 Self Declaration Procedures?
  
  Under the new procedures for DoC/Self Declaration there is no longer a
  need to submit/certify computing devices and peripherals with the FCC.
  
  The manufacture simply has the testing performed at a A2LA approved
  OATS, applies the new FCC label to there product, and issues a
  Declaration of Conformity to be shipped with the product. The real
  benefit is that there is longer a submittal to the FCC, so a product
  can hit the market as soon as it is shown to be in compliance, saving
  several weeks in the overall process.


I'm in the UK and I read this as meaning you just can't rely on any CE technical documents for short-cutting the procedure in the US especially for EMI. Furthermore, the same link clarifies this a bit about testing: -


  Differences in emissions limits?
  
  I have heard that the emissions requirements for Europe, Japan, the US
  and several other countries are all very similar.  Is it possible to
  perform one test that covers several markets for emissions?
  
  Depending on the type of product that you manufacture, it may be
  possible to perform one emissions test, generally based on the CISPR
  22 limits, and only perform deviations for the other markets.
  
  However, it is important to confirm that your test lab is qualified
  and accredited to perform testing for these different markets. For
  example, the VCCI (Japan) requires that the testing laboratory be a
  member, and if you are going to self declare a computing device to
  meet FCC Part 15 requirements, the lab must be A2LA accredited. PSE
  holds both of these accreditations.
  
  Keep in mind that whether you are trying to meet EMC or safety
  standards, it is always important to define all intended markets
  before testing begins. PSE always strives to meet our client's needs
  in the most cost effective and time efficient course possible. By
  having a clear understanding of which markets you are seeking, we will
  be in a better position to satisfy your needs.


The key point is that if the UK or overseas (outside the US) lab has A2LA accreditation then this makes it an easier exercise.