Just realised at #4 the OP references outlets, this inferrs a ring or radial circuit with s.o.
I guess 13A type.
If the compressor is on its own dedicated circuit which unless domestic it really should be, minmise inconvenience in the event of a fault, and a host of others...
IF it is pugged into a ring, then other "base" loads will have an effect on the breaker tripping, and, a C or D type may not be appropriate, it should be carefully considered.
Also, if not on a dedicated radial, the FFS! A 1 pole D16 is a tenner list! IF it is on a dedicated radial, then the original design was insufficient to allow for the inrush. Still D16 3pole around