How many holes? It's a massive structure and doubt a couple or few holes would pose much of a warpage threat. How flat is it to begin with?Some preheating would help but again hard to get much heat into it.
Copper works but it can still stick, especially if it's a plug versus backing plate and after a few cycles. Think I'd just shove a drill bit through the hole so you had clean fresh metal and press/pound a shallow steel plug in the bottom, then fill it up in a circular motion with high feed and power. The bottom of the hole will never be a problem to drill through because it will always be attached via some weld.
......drilling through a (hard) weld in the future may be another matter.
Best,
Kelly