Your first problem is that you only have 1 weekend per month. That is very unusual, and indicates there is either some geographic distance involved, or some major issue in your relationship with your children. Your second problem is that you are focusing on your ex instead of focusing on your children. Do you really want to tell your children they can't play in their games, perform at their recitals, or otherwise engage in their extracurricular activities because your time with them is more important? What will you actually be doing with them that is more important? Why isn't spending time watching them play, perform, etc. equally important? Now if your ex is saying you can't go to these activities with the children, that is a problem. And certainly some things may be frivolous. But overall, your children may very much come to despise their time with you, if that time means they have to give up activities they love. On the other hand, your ex should not be signing the children up for anything without at least consulting you first (assuming you share joint legal custody). Many court orders have language requiring that parents agree on activities, and if they don't agree, the activity cannot impact the non-consenting parents time. But I have also prepared and seen court orders where a non-cooperative parent is required to get the children to all activities on his/her parenting time. Overall, unless your court orders say otherwise, you can do what you wish on your time, and you do not have to allow the children to participate in any activities you did not specifically agree to.
Responses are for general information purposes only, and are based on the extremely limited facts given. A consultation with an attorney experienced in the area of law(s) indicated in the question is highly recommended. Information and advice given here should not be relied upon for any final action or decision, as the information is limited by its nature to the question asked and the fact(s) presented in that question. THIS RESPONSE DOES NOT CREATE AN ATTORNEY/CLIENT RELATIONSHIP, particularly considering that the names of the parties are unknown.
Each parent's respective custodial time belongs to that parent. Accordingly, any activities scheduled by the other parent during that time would be secondary to the parent's right of visitation. A child's time with each parent is important so activities that operate to limit (in this case it sounds like significantly limit) a parent's visitation time are not appropriate and as such, would likely not be supported by the Court. Children's activities are good too, but they should be chosen cooperatively with an eye to maximizing parental time, not limiting it. In short- you do not "have to" send the children. If she files with the Court you should seek a consultation with an experienced custody attorney to help formulate the best legal strategy to support your position in the matter. Hope this helps.
I completely agree with my colleagues here. If you can get on board with their activities, attending all that you can in support of your children, then you and your children will benefit. But you should NOT have to live your life according to the schedule your ex sets. You get to determine what you do on your time. If your children want to do what is scheduled, then you would be better off supporting them. If they are ambivalent about the activity, don't go. But I also suggest you not worry too much about her taking you back to court (unless the reason you only have one weekend a month is due to the judge's poor view of you). Nearly every judge in this district will say you get to control your time. As this is particularly fact specific, get with an attorney now so you can plan accordingly.