Last plumber- you say spillage test can vary, I know nothing about the spillage test- does the spillage test give you a yes/no or a quantified result? Ie would a spillage test let us know how close it may be to failing the test if conditions vary like you suggest?
The spillage test carried out on the day, will pass or fail and generally give the indication that a chimney / flue is working.
It is carried out with windows and doors closed. The reason for this is so that the appliances flue is tested using what we call adventitious air.
As far as cavity wall insulation is concerned, in some circumstances, filling the cavity with insulation can, ( not always I stress), block ventilation that was designed to ventilate the underfloor area of a property, ( if you have suspended floors). This could ( in some properties), reduce the adventitious air entering that room and in turn effect the flue's operation ( unlikely but not impossible). This is why there is a necessity to have the fire tested after the wall is insulated. Some properties do have solid floors and in those type of properties, cavity wall insulation will have a lot less effect on the adventitious air.
Company's that insulate cavity walls have a responsibility for your safety but in most cases they are not gas safe registered Engineers ( I suspect), therefore they will have a very generalised view of this situation and will not want to get involved with your fire.
If they insulate your wall and your fire spills products of combustion, afterwards, they could be found liable, hence their stance.
Will they not insulate the wall on the understanding that you take responsibility for having the fire tested after they finish?
Show them the manufacturers installation instructions and offer to sign something that takes responsibility away from them.
It should be tested after the wall is insulated anyway in my opinion, it is not impossible for insulation that is pumped into a cavity wall to enter the flue itself. Again, these things are all very unlikely but happen on occasion.
If they do fill the cavity wall, make sure ( please) that you get the test done by a gas safe registered engineer, so you know the fire is safe.