I use a 3/4" straight bit, I think it is called a box bit, and set the fence for 5/8". I actually find it easier to rip the rabbet with the table saw and use the router for the dado in the bottom board side rails.
also would go with 2 cuts on the table saw. If all you have is a router then get a 3/4 inch bit that is at least the height of the frame rests. then adjust the fence to get your cut. or vice versa if you run the panels standing up. The 3/4 inch size........or really the biggest you can have in your router......is for better performance. Wait you haven't assembled the box already have you?
To router rebates/rabbets in boxes which have already been assembled (special cases, certainly NOT to be recommended as a construction method), I use a small diameter (8mm -ish) router bit and make multiple passes. The advantage of the small bit is that the amount of wood left in the corners is minimal, and need not be removed with a chisel - unless you're OCD
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