Normally, an immediate after takeoff turn is for obstacle problems.
The main concerns are
(a) constraining track which is turn radius controlled .. ie speed and bank angle related
(b) for the OEI case, 15 degrees is the accepted maximum for a reasonable balance between climb gradient reduction and turn performance. As the target is V2 in the OEI case (ie the designer doesn't have much flexibility with speed) one often sees the situation where a procedure specifies a lesser bank angle to better match turn radius to the obstacle requirement
(c) unless a separate AEO case is prescribed, the AEO tracking should follow the OEI which, in turn, requires a similar speed and bank schedule. It is turn radius which is important here .. not manoeuvring.
One must not increase beyond the procedure's stated speeds as this will invalidate the assumptions which the ops engineer used in developing the procedure.
The question of manoeuvring restrictions relates only to maintaining a sensible stall speed margin in normal operational manoeuvring and is quite different to the considerations in escape planning. Imperative that the two are not confused.