Not to me it doesn't. I would think the choke would be concentric. In other words, it shouldn't matter if the choke tube is in or not. If the arbor is adjustable it should center itself on any size bore, choke or no.

Bore sighters are just a tool to save some ammo (though you can use them to record scope settings for various loads). They get you close. I guess I wouldn't worry too much about it.

I think you were asking whether to bore sight with or without the tube. I'd go with the exact setup you would hunt with.