Would having multiple plans be a problem? (Kaiser + Anthem)

I am currently on Kaiser in CA. I have an operation planned out of state with a non-kaiser specialist I found. The cost of the operation is high enough that if I switch to Anthem Blue Cross Platinum EPO plan with 0 deductible and 10% coinsurance, I would save a bundle of money.
Yes, I can switch because I recently had a qualifying event (quit my job). And yes, I've confirmed that my Anthem California plan will cover the out of state operation, since the specialist is in the greater "Blue Cross Blue Shield" network.

My question is: will it be a problem if I keep my Kaiser coverage and have two plans? I want to keep Kaiser because I like the doctors there for my other needs, and I can afford it.

Obviously Kaiser and a non-Kaiser plan are not going to have any overlap in coverage, since Kaiser is "exclusive", if that's the right word. However, would there be a situation where, for example, Anthem finds out I have Kaiser, and that they think operation could technically have been done at Kaiser, and therefore refuse to pay?

Or, I go see a specialist at Kaiser, and Kaiser finds out I could have gone to an Anthem doc, and refuse to give me benefits?

Also, is it true that, in order for me to have multiple plans, I would need my second plan to be under someone else's name? (I.E. I would have Kaiser for myself, and I would need my wife to the "primary account holder" or whatever on Anthem and I would be on it).

Any other tips/wisdom on having multiple plans is greatly appreciated.