I still don't think it's very clear from OP's posts, if the non-grammar school her child is likely to get into, is one which has the full range of ability (because the grammar is just 1 which serves a vast geographical area and selects purely based on exam results) or if the non-grammar is essentially a secondary modern with perhaps the lower 75% of ability as the upper 25% of ability have been creamed off into the grammar.
I would disregard the issue about where friends are going, especially if you live far from them and are unlikely to get into the school's they go to anyway.
I would be honest with DS about the likely options where you live and make sure he realises he's not likely to be going to any type of schools with his primary friends anyway, because of where you live. I'd be having a serious chat with him about the realistic options and tell him there's no certainty about getting into grammar anyway but his teachers think he has a chance. Point out that trying and sitting the exam leads to a choice, whereas not sitting it removes any choice.
I'd also talk about preparation involving familiarisation rather than heavy coaching. You or a tutor will need to familiarise yourself with the structure of the exam and make sure he has seen some sample questions and had a chance to do some practice of the different styles and closer to the event, had a go at working in timed conditions. It doesn't have to be intensive prep for hours a week. It could well be an hour a week from Easter or perhaps 12 weeks before the exam if that's what you felt you wanted. You can choose how much you prepare.....but it's worth knowing just how hard to get into your local Grammars are. If they are superselectives which take purely on exam and have zero catchment, then some have 2500 sitting for 80 places. Yourse unlikely to get into those who those with a little light familiarisation. However, if it's an area where 25/30% get to grammar, a bright child who has had some reasonable familiarisation can certainly get in.
FWIW I had a son who was a bit of a resister regarding 11+ prep. He found it tedious and boring (especially maths) and kept saying he wouldn't pass. I was tempted to stop the prep many times because it wasn't fun at at all, but I knew I really wanted him at the grammar. So we struggled through (about 10 months of prep with a tutor and with myself, but taking most of the school holidays off) and the he did the exam and he passed. And now I'm really pleased because he is getting the right kind of education for him. He's not top of the class but not bottom either. He's learning several languages and doing 3 separate sciences and has specialist teachers who have degrees in the subjects they are teaching. He would have been happy at the alternative school too I'm sure, but I do t think he would have done so well.
I think it's possible for most parents to coax their children towards some gentle low-level preparation. You have to go into it knowing they might not get a place anyway, but any maths and English Oreo isn't wasted anyway and the trying is the thing to praise them for, not the results.