I don't know about "unreasonable," tenants DO like to think of their home as being, well, "their home."
That aside, there's no need to worry about legal action so long as you are proceeding in a legal manner. The real worry is whether the tenant will be cooperative, and as Daniel Norden points out, you can catch more fly balls with a soft glove than with a hard bat!
Basically, give them 24 hours notice every day that it's on the market, and offer them an incentive for each business card or agent that they let in. (I just read the keybox and base the spiff on that). You want them to not only be cooperative, but to actually encourage showings!
All the best,