Don't worry. The classic Mini has been around for almost 60 years. It has been used in racing and rallying and the technology needed to run them is well established.
The only time oil has caused me a problem was in the very early days of synthetics I was given a load of Shell Gemini to try in a rally car. It damaged the bearings and syncro hubs and cost me a lot of money.
Sine then I have used first Kendall 20w50 mineral then, when that became hard to get in the UK I switched to Valvoline 20w50 and it has never been any issue, even in rally cars using high compression ratios and over 7000 rpm (up to 8000 rpm with a 970 'S' I briefly had).
I'm sure oil technology has moved on and that there are modern oils which will work well in the old A-Series engine/gearbox unit, I think I'll stick with the Valvoline VR-1 with which I've been happy for over 25 years.