Sure, as the TLGP-MYS ICE is empty for all days from ASK to MYS it would be a better option with family to reach MYS by night. Don't worry about the short time gap (45 mins) at ASK as Chalukya is generally very punctual and reaches ASK on time and YPR before time (by 20.45- 21.00) barring unforseen situations. ASK has good snacks options and stalls to munch food as well just in case.
In fact I have heard that Bus connectivity between ASK and MYS is also decent, however I have not used this option personally. Other members may share more light on this aspect as well.