Procedural question pertaining to the time line for discovery set by the court

You likely misunderstand the discovery deadlines- they are deadlines to complete, and not restrictions on commencement of discovery. A Party can serve discovery with the initial pleading. If you cannot work it out with counsel, you must file a motion to the court asking for relief. This is fairly routine for lawyers, but it can be confusing to someone in your position.

Discovery can be requested anytime up until the "discovery closed" deadline.

By District court do you mean Federal District Court? It's not typical for state district court to set any deadlines for discovery because all you get to do is serve a few written questions right at the outset and you cannot request documents except a few specifically outlined in the rule. If you are talking about federal court then discovery is a completely different animal there and you can't even serve discovery until the court issues an order telling you to go forward, and even then you are required to automatically provide certain information without the other side having to ask you a question. If the court set a discovery schedule you need to abide by it and if you don't understand what it means then you need an attorney. The court will not give you advice.