Depends on whether it's the last day of the hunt, in order to have an opportunity to get your Big One:
If it is NOT the last day of the hunt:
Yes, finish him off for sure, gut and drag back to camp. Place your tag on him. Go looking for anyone on or near the next property that appears to be walking and/or looking for their game. Hopefully, you'll find the chap who shot it, and gracefully hand it over - take your tag off, and it's ready to re-use. Go back to hunting. If you cannot find the person that shot him, then that's your buck - unless you can find someone else in camp at the end of the day that evening that wants to claim that one as theirs - then switch tags).
If it's the last day of the hunt:
Still, finish the deer off (that's the right thing to do, regardless of whatever else you do or what happens). Then continue hunting and waiting for your big boy for the rest of the day. If the big boy does not come along, then gut that deer, and take back to camp. Still, go looking for the guy who shot that one and hand if over if you can find him. If not, that's your buck. Now, if big boy comes along, shoot big boy, gut, tag, and take back to camp. That's your buck. Then go look for the hunter who shot the spike, and failing that, see if anyone in camp wants to claim the spike as theirs for their tag, and then go show him where it is and bring it back. Failing both of those, call the game warden, explain the situation, tell him that you don't think you are in the wrong because even though you shot the deer, it was already mortally wounded and you were just issuing a coup de grace. He or she will probably agree with you, and tell you that you need to either leave the deer lying for the yotes, or will issue you a special tag so you can keep it, or will ask you to go get it and bring it in to donate to hunters for the hungry, so he won't ticket you. If, on the other hand, he is a butthead and wants to ticket you, then take your medicine and pay the ticket - that's the price for getting your big boy. I might just tell a white lie as to the timing of things - say that the big buck came in first, you shot him and tagged him and then shortly afterward, here came the wounded spike, and you had no ethical choice but to issue a coup de grace.