In short, no, you should not take retaliatory action to make your point during a dispute. The courts universally take a dim view of parents that stop child support payments as a bargaining chip in a custody dispute. From a legal standpoint, child support and custody arrangements are standalone legal issues. In fact, courts can and do order child support while forbidding visitation; it is rare, but it can happen.
It 's okay to assume that as you regularly make your child support payments, you should garner good will and deserve a modified child custody order. It is entirely natural to take this position after many months and years of support payments, especially if your ex-spouse is often late or slow in transferring your child to your care.
If you want to modify custody, it is almost always better to work it out with your ex-spouse. If your spouse is reasonable, you can usually arrive at a mutually agreed plan. But if you are unable to compromise, then you can file a request for modification of visitation with the court.
You can also request joint physical custody. However, the court will only order this if it is in the best interests of the child.
If you are engaged in a custody dispute, then you may want to speak with an attorney. Modifying a child custody arrangement is a delicate scenario. You need to frame your arguments from the perspective of what is best for the child. It is very easy to fall into what you believe if fair for you. But it is unwise to approach a modification order from that standpoint. A lawyer can help you frame your arguments to ensure that you maximize your chance to receive a favorably modified agreement.