A good trundle wheel is accurate to about 1% so has a possible error of a little over 6m in the distance you are talking about. A GIS measurement would be far more accurate than that. A GIS system would typically measure in straight lines running up the centre of every road/footpath used and with no cutting of corners. That would usually produce a longer measurement than you would get using a trundle wheel. There is also the question of precisely where on your property and the school's property they use as the end points for the measurement - if, for example, you are measuring from your front gate but they measure from the centre of your property that would add a few metres to their measurement.
The LA may well specify in their admission criteria that the distance is as measured by their GIS system. If they do, their GIS system trumps any alternative form of measurement. You will, in any case, have a lot of difficulty persuading an appeal panel that a trundle wheel produces a better measurement than the GIS system.
I'm intrigued by your comment that "they have used some straight lines in their calculations but that isn't the shortest route". Do you mean that their route measurement doesn't follow the roads/footpaths or that you have identified a shorter route than the one the LA has used? Or do you mean something else?
I am very surprised that the LA hasn't allocated a school for you. In general they are required to find a place for your child. There is a kind of "get out of jail free" card in the regulations in that if there are insufficient places to go round they are only required to ensure that they leave the minimum number of children without the offer of a place. However, leaving 421 children without places is excessive - that means they need another 15 reception classes. If there were 300 children without places last year they should have known that they were likely to have a problem again. In my view it simply isn't good enough to leave you to work it out for yourself. With that many children without places there is no chance that you could find a place for your child. They are basically hoping that people will go private, move out of the area or home educate. Not acceptable.
If they only added the extra class last year that makes it less useful. It may be that the school only had enough capacity for an additional class for one year only. They presumably now have 3 classes in Year 1 instead of the normal 2. There may not be enough classrooms for 3 Reception classes as well. It is worth finding out whether the school actually could handle an additional reception class. However, I would still pursue it, especially if your councillor can back up what the parliamentary candidate said - the parliamentary candidate may have been speculating rather than stating fact but the councillor should be able to determine the truth. I would also bring up the serious shortage of places in the LA and the fact that you haven't been offered a place anywhere. Strictly speaking it doesn't give the panel a reason to admit your child but it does highlight that the LA has failed to meet its responsibilities.
You should also be asking the LA what they are going to do about finding a school for your son. If you don't get a satisfactory answer, make a complaint in writing (email will do) to the Admissions Manager and the Director of Children's Services.
You don't sound at all shrill. I'm not surprised you are worried. I hope you get a good result, either through the appeal or by the LA pulling their fingers out and sorting things out.