If at all possible, under the circumstances, I would have left it standing.
I've been asked by foresters and loggers alike, in a discerning tone of voice, "Why the 'F' did you leave that tree standing?" In most cases, after explaining, I was dismissed of reprimand.
Works almost every time with bees. As for seasonal nesting trees we worked around many and came in after the fledge and finished up the job. You got to be a nice logger!