And the blades?This.
Have been doing it for years and never encountered any problems.
A disassembled three piece safety razor is unlikely to arouse any suspicions.
Putting the three pieces in different places within your carry-on even less so (even though I do admit that I usually place them in the same container as my electric toothbrush).
Besides, TSA and other airport security staff are generally not out to annoy and inconvenience travelers any more than their job requires. A bit of common courtesy and non-confrontational attitude goes a long way.
B.