Since what you wrote was rather ambiguous, let me clarify how I'm understanding you: I presume all letters are PoW transactions, and arrows show inheritance. So by a double spend, you must mean a double claim of transaction fees (from the otherwise burnt half), since without access to my private keys, you obviously can't spend my outputs (and PoW blocks do not have inputs in any case).

As ever, the network selects the TDAG (in this case, branch) with higher total transaction fees paid. People making PoW transactions have very strong incentives to pay seriously high fees (e.g. 25%) because otherwise they risk their PoW transaction being replaced by another with the same parents but which pays higher fees. (Two people finding a solution to the PoW puzzle at roughly the same time is not that uncommon.)

So, if you are paying sufficiently high fees, the network may switch to your TDAG. But if mine is well enough embedded in other transactions (paying fees), then even if you pay 100% fees, you might not be able to produce a switch.

In any case, it seems that all you're describing is a classic 51% attack.