The document is the generic "extending your home" planning document which speaks of subservient extensions and I can see some of the logic but not the width reduction.
Every local authority has different documents, whilst many of the policies are similar they are not the same.
I was going to use the environmental costs of manufacturing, delivering and installing large RSJ's and sending most of the garage floor to landfill as part of the arguement.
This won't change their opinion and will be a wasted exercise. (If the environmental argument stood then the most environmentally friendly thing to do would not build an extension).
Usually just a front set back and lower ridge is sufficient when building over a single garage.
How wide is your main house? As a rule of thumb usually half is easily acceptable (i.e. if your house is 8m then 4m would be fine), but wider is also often fine too but more subject to design.
The property across the street has been extended to similar proportions so why not ours.
Yes, every single property is totally unique as plots were purchased on the street over many years so we've got every variety of building going. Some have two plot widths etc..
Not sure how the building opposite can be totally unique but have done similar to what you require.
If every house is completely different then the subservient design requirement makes no sense. This is typically the requirement when there is a consistent pattern of development and so extensions shouldn't break this, keeping the reading/legibility of the existing houses intact.
If there are larger houses in the street than yours would be after development and every house is completely different already then the subservient argument is probably flawed as the reading of the original house isn't really important.
From a design perspective if the end results isn't out of scale to the context and looks good it should be acceptable subject to meeting other policy requirements.
Another design approach might be to unify the extension with the house (if possible) so it looks like it was all done in one go, rather than an add-on. As above if it doesn't match anything at the moment then It would be no more not matching afterwards.
Ultimately if planning are unwilling to negotiate you probably need a planning consultant and be prepared for a refusal and a planning appeal.