Small picture comments:
You use two variables to identify a config item/ section and name/.
This increases the code size...
The user of the library can implement their own config structure in the name with punctuation/ for free.
The following is a pretty expensive way of getting the next highest power of 2; it is inline and maybe GCC will convert it to a constant load - if not, there might be better ways for a micro-controller with no 16 bit multiply..
bytes = 0x01 << (uint16_t)ceil( log2( Size() ) )
You allocate the value buffer on the heap, inside the constructor, which will inflate the memory required by a heap prefix/postfix chunk.
You could use the template to create the value as a member variable.
Then if the object is a static variable of the sketch the minimum memory used, and if it is an auto variable on heap, it is all on the stack.
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Really big picture comments 
First, My comments come from reading your code, and I hope that they are useful to you. You have done a lot of work that I have not commented on, as it seems completely reasonable and rational to me;
And your code is clear and easy to read which is a great thing; other-wise I would not be able to make these comments.
However, an important thing that is missing in the package is a short statement of objective.
I ** think ** it might be something like this...
This Config package provides the ability to store configuration values identified by name in the EEPROM of an Ardunio.
The configuration will be accessible by all sketches that run on the Arduino - so multipe versions of a sketch can share configuration data.
Variables can be read by sketches without needing to know the datatypes [I think you implemented this in your code - not sure].
Configuration variables can be treated like very short files...
And if the above is the objective, a simple sketch that dumps the contents of the configuration variables to the serial output would be a good test tool.
And a more complex sketch that allows the deletion/insertion by hand of config values would be useful.
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To restate my comment from above/ your library is well written and very clear.
You could make the code more understandable for others now, and yourself in a few years time, by including short 'statements of intent' in the code before each larger section.
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I hope that these comments are of assistance to you.
Dafid