Did the paed talk about referring you on to others? It sounds like they haven't done a dx but have acknowledged that other professionals need to be involved. I think going back to the GP at this point might be unnecessary unless the paed is for some reason not referring on for a formal dx.
The way it works round here is that GP refers you to a paed and the paed does an assessment to see if there are issues and then refers on to the people who can make the diasgnosis - the clinical psych or psychiatrist and usually a SALT and an OT and maybe others (DS had his hearing tested for example to make sure his behaviour was the result of not being able to hear). It maybe with the SALT appointment already lined up that you are on to that bit now.
After that, when everybody had seen him, the paed, clinical psych, SALT and OT got together at the end, with us and the SENCO from the school (DS was at school by then but was in nursery at the start of the process) and came up with the AS diagnosis.
Everywhere is a little bit different but you need to check whether the paed was initiating a team dx or whether some how or another you have hit a dead end and they aren't actually going to refer you on to the people who make a decision. I think I would phone the paeds office if I were you, and see what is happening as a result of the report you have.
My personal opinion is that it would be better to know for sure what you are dealing with before embarking on statementing but that is just my opinion. In theory you don't need to have a dx to get a statement but it should be more harm than good to have one. I agree you need to focus on one thing or another though.
I wish you all the best - it all seems horribly confusing in the beginning but it should sort itself out eventually if everybody is dong their job. 
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