V2 min is only 1.15 or 1.2 Vs (depending on aircraft certification). Coupled with the other problems brought on by unexpected engine failure, this is not much of a margin above stall, so is probably the main reason for the 15 degree bank angle limit.
If a V2 overspeed is permitted ('improved climb'), although the close proximity to stall is then not such an issue, the takeoff analysis should still assume any turn below 1500 feet will be at 15 degrees bank angle and take into account the greater turn radius.
Some SIDs which require a low level turn after take off may state minimum bank angle greater than 15 degrees. V2 should be increased in the analysis to cover this.