Hi, this is a pretty tricky problem in general and I don’t think there’s a consensus on post-hoc ADVI diagnostics, but you may find the following paper somewhat helpful. Yuling Yao gave a presentation of his work on PSIS diagnostics at StanCon helsinki and it seems quite promising, but your mileage may vary.
[Yes, but Did It Work?: Evaluating Variational Inference] https://arxiv.org/abs/1802.02538
Yuling Yao, Aki Vehtari, Daniel Simpson, Andrew Gelman