I don't fly a B737, but we would commence acceleration at the acceleration altitude of 3000' and retract flap on schedule. Once clean we would continue acceleration to 250 knots. The question about staying slow for the turn would probably rely a bit on local knowledge. In my experience if a speed slower than 250 knots is required to achieve the turn radius depicted on a chart the lower speed is specified, e.g, "max speed 220 knots in the turn". Airmanship would normally dictate that unless there is a specific reason not to, you should get to 250 knots as promptly as your procedures allow so that the departure controller is dealing with aircraft all travelling at a known speed. In Australia it is a requirement to commence acceleration to 250 knots at 3000' AGL so that the controllers know what everyone will be doing.
I'm not really sure why you are looking at the immediate turn after take-off procedure. An immediate turn would be one where it says something like, "as soon as practical after take-off, turn left, track direct to GONZO". A turn at 3.1 DME is not even close to an immediate turn after take-off.