I tries that with the exact same results... Attempted mmap_anon of size 200MB...fails...Steam fails, with ENOMEM.
Here's the exact mmap call... it's the only one that fails, and it's the largest without MAP_NORESERVE.
mmap2(NULL, 209780736, PROT_NONE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) = -1 ENOMEM (Cannot allocate memory)
Running the same call from a test program suceeds - I can mmap() things that are an order of magnitude larger without problems (as a regular user).