I've cut 2x4's and their cousins many times on my SB and always use a regular run of the mill 1/4" upcut Onsrud (or similar) with a 1" LOC to do it. There is no need for anything fancy here. 2x stock is simple, soft pine. It cuts very easily. As long as the collet clears the wood, the bit can be used. I've many times dove below the cutting part of the bit without a hitch. I typically will use a 6 IPS and 14,000 RPM with .1" depth for this type of cut. If you don't see a plume of chips, you're probably going to have to get out the shopvac and lend a hand but the cut will still go ok.
If you're not cutting a hardwood like maple or oak, you can get away with a lot of shortcuts.
ShopBot Details:
2013 PRS 96x60x12 (Centroid upgrade)
4hp Spindle
12" indexer
Aspire
Rhino
Fusion 360
Ferrari 360
Prusa MK3S+
Prusa XL multi-tool