Things that imply BQP Derandomization
Interesting!  I don't know of any nontrivial result of that kind -- neither one that says "if [XYZ] then BPP=BQP", nor one that says "if [XYZ] then BQP &sube; BPTIME(2o(n))."  The closest I can think of is a result of Terhal and DiVincenzo, who showed that if constant-depth quantum sampling can be efficiently classically simulated then BQP &sube; AM.  (Later work, by Bremner-Jozsa-Shepherd and myself and Arkhipov, shows that the same assumption would even imply P#P = BPPNP.)  Combining that with some assumption that gave AM &sube; BPTIME(2o(n)) would then yield what you wanted.