I have been in a similar situation before, except that I offered the bounty as a result of an exceptional answer even though it did not answer my question. It was a rather technical question and at the time the reason for it happening was not very clear (it turned out to be a bug in an oracle library (link to question if you were curious).
At the time I did not mark the answer as correct, it took several months to get to the bottom of the issue. However, the effort put in by the person who answered helped so much that I felt it was deserved to award a bounty. Without their help the problem would have definitely taken more time to solve.
In the end, it is the checkmark which indicates to people what solved the question or issue at hand. The bounty is available for so many reasons that it does not necessarily have any one strong indication when awarded.
All that being said, there is one situation that I think is important to keep in mind. If an answer posted somehow poses a disadvantage in the future and a bounty is awarded to that answer, then it is important to leave a clarifying comment so that future visitors do not use the disadvantaged portion.
Ethically, I see no downside to award credit to someone who put in a very good effort and helped to at least narrow the problem down.