I believe Bucks will only do a remark on request (and starting next year I hear they will charge Â£10 a time!), so when I started reading your post, I thought you were going to praise the other borough for their speed, initiative, and zero cost!
Technically, I would have thought the test papers are the "property" of Admissions, and they're at liberty to re-mark them if they wish. I've never come across an "independent re-mark", but I do know that test papers are exempt from the Data Protection Act and that you have no right of access to them.
The results of reasoning tests can never be 100% reliable. For example, with a score of 137 (the 99th percentile) there is a 90% chance that the "true score" lies in the range 124-140.
I understand your concerns, but that, of course, is why there's an appeals process - so that you can argue your case in front of an independent panel.