I am in a similar position as you and I have asked this before (see link 1). Studying only mathematics for a year is not necessary, but it's up to you to determine how comfortable you are wuth math. Many TCS papers contain a lot of math, which can be daunting, especially if they are not from your area of expertise. I am learning as I go for more than half a year now and I have accumulated more knowledge than I had in the past two years studying. TCS.SE has been a very good stimuli, since I am exposed to problems that I normally wouldn't, unless I attended a dozen of conferences or so.
However, studying only mathematics for a year can be dangerous, too. First of all, you could be intimidated if you allow yourself to go too deep in a subject. Although some undestanding is required, some areas of mathematics are supposed to be only a tool by computer scientists. A certain level of knowledge is needed to effectively use the tool, but you don't have to know how exactly it works or how it would be improved. To reflect this, reverse your situation and assume you are a mathematician needing some programming skills. The amount of knowledge from TCS would be minimal, yet going too deep in TCS could lead to abandon your goal altogether.
The previous was a "DFS" kind of problem, i.e. going too deep. The other problem is expanding too much , i.e. BFS. There are many areas of mathematics with complex relations between them, sometimes even being incompatible. Fields are becoming more or less relevant as the knowledge of mathematics progresses and new breakthroughs arise. Simply put, you could be caught on the crazy new trend in mathematics, which can overstate or understate the importance of certain areas. That can be good if you are a mathematician, however it can serve as a distraction if you are a computer scientist. Some areas of mathematics may be irrelevant to your cause altogether. For a more hands-on example, consider a mechanical engineer. His machines follow the rules of quantum physics, but he doesn't care about that level of abstraction, he can work using classical mechanics and ignore the quantum level. Yet, it is possible that if he studies physics in detail quantum mechanics will stray him afar from his initial goals.
All that said, although "getting lost" in mathematical knowledge is dangerous, going a bit further than needed is beneficial. This extra knowledge can help you see patterns and solutions that you wouldn't otherwise , expanding your "arsenal" with new thinking approaches. As you can see, at least in my opinion, knowing how far is far enough in terms of mathematical knowledge is not an easy problem (how ironic). That is why the "learn as you go" method is preferred, since you will have experienced computer scientists at your disposal to guide you. Your advisor and tutors can help you determine what you really need to know and when your mathematical studies get in the way of the computer science ones.