I find that I may need a case larger than my usual 28"suitcase this trip to RSA on SAA. IF I look at the 30" or 32" cases which would give me the length that I need, they measure slightly higher by an inch or so on the posted maximum measurements. I am flying business class, so not at all worried about the weight.
What can you all tell me about how much wiggle room and accommodation the airline gives (business) travelers as far as classifying and charging for "oversized " luggage.

Read the airline you are using’s website. Each is a rule unto itself.

One common exemption is sporting implements. Gun cases qualify.

That being said, it will often be up to the individual agent as to how they implement it. You may well get by on the way out, and get dinged on the way back... it’s not guaranteed. Business is a little more lienient than economy, but not always...

I think it depends on the Airline also.

South African Airlines
Checked baggage dimensions

Please make sure your checked bags are within the dimensions outlined in the diagram. Bags greater than the dimensions provided are too large for airport conveyor belts and will not be accepted on the flight.

158 CM = 62.2 inches

I looked on line and found A couple of different bags and there dimensions.

30" + 10" + 18.5" = 58.5 inches
30" + 17" + 15" = 62 inches