Physics not Finance
The better comparison for SOU is LCY which routinely loses multiple flights each year through fog with many delays, cancellations and diversions. CAA stats tell a sorry tale.
The constraint issue at SOU and LCY is similar, the runway length and the local terrain and obstruction characteristics (hills & buildings) prevent use of more sophisticated precision approach aids. This is primarilly a physical not a financial constraint.
The proportion of flights cancelled at SOU due fog over a year is tiny. The cost to the airport is very small.
Any investment made at the airport would far better go to a parallel taxiway and more stands, than improved approach aids. This would partly address the bigger business issue; south of England airport capacity during summer.
However, investment near term is highly unlikely. Glasgow not SOU is the jewel in the airport owners crown.
FF