The testing range of the technique is the limits between which the results are shown to have acceptable accuracy and repeatability. The accelerator mass spectrometer will have a limit to the concentration at which is can accurately quantitate the amount of carbon-14 (currently between femtomole and attomole levels of carbon-14). This currently equates to around the 50K year range but can be improved on in specially designed laboratories which are capable of reducing the background carbon-14, for example. It is likely in future that more sensitive instruments will be designed that can extend the range further.
The problem is you are trying to fit the text into your beliefs. You need to let the text speak for itself. Look at the literature - the repetition, rhythm, the number symbolism, the chiastic structure, rhyming words, etc. Look at the structure of the text - the two triads - one where the domains are created and the other where they are filled. Stop trying to force the text into a linear chronological structure which is a very modern, western way of thinking. See the cosmology as it fit into the ancient world. This is what the ancient Hebrews actually believed and the three-tiered structure is seen right through the Scriptures. Stop trying to separate the 'natural' from the 'supernatural'. Again this is a very modern western way of thinking.
Whether you think the material came from Adam or not, you have to admit that Adam did not write it down. Therefore you have to allow the author of Genesis to have written it in their own way to convey their own message under the inspiration of God.
Put the text in its proper context.
The RATE project has many, many problems.