The best advise is to see a lawyer. IN family law, like other areas of law, you gets what you pays for. So get a good lawyer. In some provinces there may also be a family law legal clinic that could offer you some free basic advise.
This link is for information in British Columbia, but the situation is pretty much the same in other provinces.
http://www.bcfamilylaw.ca/custody.html
On custody the important paragraphs may be these. I've bolded a couple of parts.
"Factors Considered in making Canadian child custody and access awards:
In making a custody, guardianship or access order,
the Court must look to the best interests of the child.
In making a custody order, the court takes into consideration the condition, means, needs and other circumstances of the child. Relevant issues are the health and emotional well-being of the child including any needs for care and treatment, the love and affection between the child and other persons, the education and training of the child, the capacity (including financial) of each prospective caregiver to look after the child, and, if appropriate, the views of the child (typically more important as the child gets older).
The Court will consider who looked after the child while the parents lived together, what each parent's plan for the care of the child is following the separation, the degree of bonding between the child and the prospective caregiver, as well as the amount of time that the parent has to spend with the child.
A child’s race, culture or aboriginal heritage and the custodial parent’s willingness to respect and foster the child’s cultural identity is a relevant consideration.
In Canada, the conduct of one or both parents is not considered relevant to the determination of child custody unless the conduct is relevant to the ability of the person to parent a child. Personality, character, and stability will be taken into account. A person’s alcoholism, drug addiction, sexual misconduct, dishonesty, and lack of social responsibility may also be considered in determining the best interests of the child.
The willingness of each parent to allow or facilitate access to the child by the other parent is also an important factor. The court must seek to make an order that will facilitate the child having the maximum contact with each parent, subject to the child’s best interests.
The Court is typically slow to change the status quo where the children are happy and in a stable setting. However, if the long term best interests of the child require a change, the court may so order notwithstanding the immediate discomfort and emotional upset. "
I would think that given the conservatism of most judges, they would take a dim view of the mother being a prostitute. AT the same time, if it can be shown that the child is happy, protected, well take care of and not exposed to the prostitution, then it might not be a problem. So, for example, if you were doing in calls out of the family home ... that wouldn't be good. If you only did calls during the day when the kids were at school or otherwise cared for, that would be good. If you had a separate incall so the kids were never exposed to the business ... that would be very good.
But like I said ... go see a real lawyer.
Tom