the safest bet for us is to say the Burke wins because it has more VLS, but then again once both ships fire dozens of supersonic cruise missiles at each other if they have no support from an AEW&C then most likely both would have very little reaction time resulting in both ships sinking each other.
the ambiguity of whose missiles are better is what makes ship vs ship battles rare so the prefered way to sink a ship is with aircrafts & submarines, if China wants to sink a Burke they will use an H-6K or another aircraft with a cruise missile fired outside the air defence range of the Burke, same thing goes for the US, if they want to sink a type 052D they will use an aircraft firing a missile outside the defence range of the 052D or may be they would use an SSN.
True on the last paragraph. However both ships can still shoot down whatever antiship missiles that are sent against them. The Burke has better stamina however, having more VLS, but ship to ship only engagement it only has 8 Harpoons on the Flight I and II and no antiship missiles with Flight IIa. Burke plans to get better antiship complement in the future (antiship capable SM-6 and Tomahawk, NSM and LRASM). 052D' U-VLS has more potential for ASM development due to its size and can get quad pack SAM in the future. But currently YJ-18 > Harpoons, and the 052D might carry 16 of them from the two banks nearest the CIWS and the bridge.
On the first paragraph, the Burke doesn't have supersonic cruise missiles. If both ships are in light of sight of each other, they will throw SAMs at each other. Once again the Burke has more SAMs, but the 052D has bigger SAMs. The HQ-9 missile has a warhead as big as a YJ-83. Both ships have extensive electronic warfare, decoys and CIWS.
It boils down to that odd missile that will get through.