JMB is the lug drilling expert, but coming from an engineering background myself I can vouch for using the high speed low feed approach to stainless steel. I would use a drilling and tapping fluid rather than oil as oil tends to heat up. Beginners including me when I was an apprentice tend to put pressure on the drill to force it through. This is pretty much guaranteed to screw up. Let the cutting edge do the work and stop at the first sign of heat build up. Stainless is tough, buy the best drill bits you can get.
As to engineering firms, spark erosion and welding. Well it's only metal and can be repaired, but cost is a factor as a cartel watch is only just over a hundred Bucks. Maybe just take the loss as a learning experience. Worse and more costly mistakes have been made and you have some useful spare parts now.