I have a logitech V470 bt mouse that has always fallen asleep when I walk up and want to use it. I must click a button and I get a picture of a mouse on my screen showing that it has once again started working. I like this. It lets me know it's working. I don't think I've ever seen the dialog, of if I have it's been so long ago I don't remember. Perhaps back when I had a blackberry and let it pair with my Mac. I'm just not sure.
I have the same mouse on my work issued win 7 pc. Whenever I boot up or come back on from standby, I have to look at a rather large box telling me my bt is not discoverable. Well duh. No crap. If I want it discoverable, I'll make it discoverable. I bet the only way to shut that off is to shut off bluetooth. I wish I had some idea what the box looks like that you are complaining about. I can tell you that the one on win 7 takes up what seems like 1/8 of the screen telling me I should leave my bt discoverable all the time. What is the issue with the box you get? Is it in the way of doing something? If it is in the way, does it not go away quickly enough? I would imagine there is some preference that would allow you to shut it off but it might involve getting one of those aftermarket tweak programs like tinkertool. Macs are actually a lot more configurable than PCs, it's just that Apple doesn't put all those configs in your face all the time and every now and then you have to visit the Terminal or use 3rd party software to get something set up the way you like it.