Overhead cranes used in general industry are governed by OSHA 29CFR1910.179 and the ASME B30 series of standards.
None of these standards require the support beams (runway beams) to be marked with a capacity rating. OSHA 1910.179 requirements for capacity markings are the same as stated by 'kenvlach' above.
If the opinion of your 'inspecting body' was correct, then every industrial facility in the U.S. would be in violation!!
The only circumstance that I can imagine for that kind of requirement is in the case where you might have a portion of your runway system that is under-designed for the rated load of the crane. On very rare occasion, I've seen this where the Owner installs a crane with higher capacity and maybe does some reinforcement to the runway structure in only one area of the building. The Owner then marks the structure for the acceptable lifted loads in the various areas of the building.
Notwithstanding the above, I believe your 'inspecting body' is flat-out wrong. If they've issued you any kind of report or other document, I would press them for a revision.