Lots of kids do very little work for mocks. It's well documented.
Is it? Where? I only have my DC as a reference but they all did about the same work for mocks as they did for the actual GCSEs. When I taught in school the YP I taught usually got a GCSE grade pretty much in parity with their most recent mock. It doesn't look from your posts as tho you are basing this on much more than you older DC and friends’ DC?
most grades typically go up by at least 1-2 grades
Again, based on the DC you know, yours and others? This is not my personal experience. I think it's risky to make generalisations like this tbh. A jump from a grade 5 in the mock to a 9 in the exam in French (my subject) is extraordinary and suggests something was badly wrong with the mock marking, unless your DS missed out half a paper or something.
Sorry if some of this is not what you are interested in hearing – it's an interesting point which is why people have discussed.
To answer your initial question, when I taught in school we gave PGs based on mocks, class performance and general attitude and ability. I was usually about right. Sometimes a bit optimistic.
My own DC at a different school were given target grades and then mock grades. Not sure we ever got PGs – they applied to other schools but it wasn't an issue (non-selective state). They key thing tbh was what GCSE grades they actually achieved, and it was then that there was a bit of hustle for some.
Do you have some alternative options? If he is getting 7+ in all other subjects (yes, great grades!) in mocks without doing much work then would something else be better? A 6 in maths in Feb of year 11 does not bode well for A level tbh. As PPs say, A level maths is best suited, let us say, to the kind of student for whom GCSE maths is an easy win for an 8 or 9.
Yes I am sure there are students who gain a 6 or 7 and get an A at A level. There are probably YP who gain a 9 and then get a C in sixth form. But overall there is a clear parity from GCSE grade to A level grade in maths specifically.
Meant to add – hopefully you have other sixth form offers; for the one you have missed out on could you contact them with actual results in August? I would get in touch and ask them.