Thanks for the reply Grant,

I don't think it's that it's getting a bad plan, I think the temp statistics being auto-built on the secondary server cause a huge IO storm scanning those pages and pulling them into memory. I believe it depletes the SQL instance free memory (this SQL Server consistently has 10GB + of free SQL instance memory, with another 15GB free at the OS level) and at the same time, because of the updated statistics, causes recompiles on the procedures which then don't have enough memory to cache a new plan.

That's my best hunch from investigation.