Vipassana meditation comes naturally; Samatha meditation incredibly hard. Let me start off by saying that I can do Vipassana meditation for about 20 minutes at present. Accepting thoughts as they arise, noticing sensations without judging is relatively easy, compared to doing Samatha meditation. However, even though my body may feel relaxed and calm after doing Vipassana, my mind is still cloudy and obscured (I suffer from brain-fog and Depersonalization Disorder). Depersonalization Disorder is a sense that you don't really exist, almost as if you are outside your body, lost in a cloud. I feel calm, but my mind still feels foggy.

If I focus on the breath (Samatha), I find that I only manage to sit still for about 3-4 minutes before becoming physically agitated (to the point where my body will curl up in tension). I feel as if I need to go for a run; my mind kind of shouts at me, saying "Go, go go! Get up. Go do something". There is no anxiety, just physical and mental agitation. 

My mind feels fine doing Vipassana, jumping from moment to moment, letting things be. But when I tell the mind to focus single-pointedly, all hell breaks lose. 

Should I carry on with the breath, even though I can only do it minutes at a time, or should I do something entirely different?
I have also found this to be the case when practicing samatha meditation. Here is my understanding of why that is, and how to deal with it:

In vipassana meditation, you do not need to "tune" sensations out. In fact, noticing them is part and parcel of the practice itself. Restlessness and agitation are thus sensations to be noticed and analyzed, and their presence won't really pose a problem. 

By contrast, samatha meditation is about concentration on one thing -- in the case of anapanasatti, the breath. One thus becomes far more sensitive to the hindrances that obscure the goal of single-pointed focus, and they actually become an obstacle. The agitation you describe is literally the fourth hindrance, which is described as an inability to calm the mind. See https://en.wikipedia.org/wiki/Five_hindrances#Restlessness-worry_.28uddhacca-kukkucca.29. 

On a deeper level, agitation / restlessness are manifestations of a mistaken belief that changing to some new state will end an unpleasant feeling. You thus experience the unpleasantness  of the sensation and react by attempting to change the situation.    

How to deal with it: 
Whenever I practice samatha meditation, I invariably experience what you've described above. To overcome it, I walk through a counting exercise I learned in Bhante Gunaranta's book "Beyond Mindfulness in Plain English". He recommends counting each in-out breath up to ten, then back down to one, then up to nine, then back down to one, then up to eight, etc. etc. until you reach one. Oh, and if you mess up, you need to re-establish your mindfulness, and start over :D 

From personal experience, I've found that this method works very, very well. I am an inexperienced, impatient meditator, but this method -- when applied with mindfulness and self-honesty -- completely subdues the grosser manifestation of the hindrances for me.   

Additional suggestions: Here is where your vipassana practice can and should bolster your samatha practice. Recognize the sensation objectively; recognize its impermanent nature, recognize that moving / shifting / getting up / etc won't truly eliminate the suffering within the experience, and recognize that there is not really a self behind the unpleasant experience -- it is merely an aggregation of perception resulting in you experiencing a hindrance to meditation. Through this process, you will calm the unpleasantness of the experience, and the hindrance will become subdued. Then you gently re-focus and proceed with the breath.