I don't think anyone can predict how the bureaucrats will deal with this. It shouldn't be a problem. Will it be? Hard to tell. I know that isn't the answer you are looking for, but nobody can tell you whether it will have an affect.
Name, DOB,and social together form the basis for your being identified by the government you can expect some issues. Recommend you contact DFAS, VA, IRS and CA state before you start the process.
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Contact DFAS they would be better able to answer.
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Anecdotally, I have seen a number of people go through name changes because of marriage or divorce in both Active Duty and Retired without much difficulty.
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It should not be too big a deal. I would recommend you contact the Defense Finance Accounting Service (DFAS) Retirement Pay Division.
Defense Finance and Accounting Service
U.S. Military Retirement Pay
PO Box 7130
London KY 40742-7130
Toll free 1-800-321-1080
I believe that as long as your SSN stays intact, it should not be a big issue. I would imagine that the retirement pay would continue to flow to your account.
However, I would keep all legal records of the name change so that if/when you pass away, whomever you designated as your beneficiary for the Survivor Benefit Program (SBP) will still be entitled to receive your pension.