If the original poster (OP) has had the child over six months, then I would think that he has an good chance of formally being named the custodial parent, if he has not already been so named. If he was good enough to be the custodial parent way back then, what has changed in the situation to warrant a change in custody?
A family court is not going to enforce their private "out of court" agreement, even if it was notarized. If that is what she is alleging in her petition, she likely is going to lose. Looks like she does not have an attorney, if that is what she is saying in her filing. A family court might recommend that the parties come up with some kind of schedule for phone time or a mediator might recommend same. Absent such a schedule currently being incorporated into a court order, there are no contempt of court sanctions to impose on the OP for violating a court order that does not even exist at this point.
What I would suggest to the OP is that he consult an attorney and ensure that he is named the CP. She gets standard long distance visitation and since she created the distance, she bears the costs of travel of the child. She may well get joint legal custody.
On the matter of phone time, I recommend that OP propose a schedule for same. Few judges are going to agree to daily phone time since it is so limiting in other ways. But, every Wednesday at 7:30 PM and every Saturday morning at 9:30 am (unless the child is with the NCP) might seem reasonable. Then the OP or his parents, if the child is in their care, must make the child available when she calls as scheduled. When the parents cannot agree on something like this, what family courts generally do is order a schedule.
As far as the child being in the care of his parents when his job requires him to travel, I would simply note that she apparently is employed (or in school) and likely would have to place the child in the care of someone else during some of those times as well. Just because he is the custodial parent, does not mean that he cannot make alternate child care arrangements from time to time. Obviously, it is one thing if he is on the road 90% of the time, but quite another if he is out of town for two or three days a month.
Soon, the child will going to school nine months of the year, (if she is not already in kindergarten). Might be a good idea ot think about what the visitation and phone schedule should be once she is in school on a regular basis and incorporate into future orders what that should look like.