I'm about to route tenons to attach a breadboard end on a table top. I obviously don't want to screw this up, as I have spent a lot of time on gluing up the top (with fair results), and don't want to spoil the whole thing. I'm also looking to expand my tool collection with the addition of some hand planes. Since this project is on the front burner, I might as well buy the correct hand plane for fine tuning tenons to match the mortise. What type of plane works best here?
I was thinking of a rabbet plane Do the blades on these run the full width of the iron? so as not to leave a lip on the inside corner of the tenon? If so, wouldn't such a plane work well for this? Do they work well on cross grain work?