Assuming your RPA is plotted as a circle centred around the tree, your RPA has a radius of 12m from the stem, right?
Therefore, if the proposed extension encroaches 3m inside the RPA, there is still a 9m distance between the tree and the extension.
Whether or not breaking ground to investigate the presence of roots is needed will depend on the site conditions - if the ground conditions are broadly similar all around the tree then it would be sensible to assume, using the BS RPA calculations that roots will be present. That far from the tree they are most likely to be small diameter/fibrous roots rather than anything of a size that would affect the anchorage of the tree (although fibrous roots do assist with anchorage).
An air-spade excavation along the line of the proposed footprint would quickly tell you what quantity and size of roots are present - with the least possible damage to roots assuming the tool is used correctly. However, given the expense of such equipment with operators, a careful hand-dig should suffice. I would have thought a depth of 30cm will tell you enough, but bear in mind that soil conditions play a major factor in rooting depth. In a compacted soil roots tend to head towards the surface where water and air are most readily available but in a loose texture free draining soil they will often be deeper.
Other options to think about - could a mini pile and voided raft system be used for the foundations so that root damage is minimised? Is it justifiable to alter the shape of the RPA of the tree? Is the soil a shrinkable clay - how may this affect foundation depth? If it is a shrinkable clay this is another reason to use a pile and raft system....