It is better to create a separate signed transaction to move funds from stuck address to your target new address AND bitcoin address of the "tx pusher". In case of success you will have your funds on your new address, and the "tx pusher" will receive his bounty to his bitcoin address (as both outputs will be in the same transaction).
That's a bad idea. It's likely these funds will be stuck for many more years, and each signed transaction will remain equally valid. Once a signed transaction is out there, it can't be invalidated until OP manages to move his funds.
Besides, it will mean trying to double spend on top of the current problem.
even if this method was desired a better bitcoin-ier way would be to use the cryptography option that bitcoin already offers.
the goal is to make two outputs, one to the owner and one to the one who claims the reward. that is the very definition of SIGHASH_SINGLE. you sign the input and the corresponding output of it (the fist output since there is only one input) and set the amount of it to the (total input value - tx fee - reward) then the person claiming the reward simply adds the new output equal to (reward) amount and then submits it to the block.