LoG filter creating additional maxima in scale space
Without exactly seeing your code it is hard to know for certain, but I suspect this is because you are not tracking your maxima well through scale space.

The position of the maxima will move in scale space. At the smallest scales you get the 4 small blobs around each circle. These are due to digitization and possibly noise/irregularities in the boundary edge. 

These maxima will expand as the scale increases and merge together to form the "correct" blobs around the circles.

However, the other side of the peaks will also merge together resulting in the 12 medium size blobs and finally these will merge to form the four largest blobs. 

Note at each stage here the number of maxima decreases 36->21 (9+12) -> 13 (9+4)

The problem is as you only look for local maxima in scale-space you lose  a lot of the relationship between the maxima.

One option may be to look for all local maxima at each scale and track these through scales. However, I'm not sure how straightforward this would be or how you determine which are the significant peaks using this approach.

A possibly better option may be similar to what you suggested by thresholding the responses. This article does basically the same thing as you. It applies a threshold on the mean blob volume over all scales. Which it claims it better than thresholding the peak response.