I think one thing you need to look at is what the risk is.
Is it 1 time in 10 in which case it's a huge risk and I'd avoid it.
Or 1 time in 10 000 000, which is low risk.
I do tend to take America's medicine/research with a little pinch of salt because so much is guided by money rather than facts. Within one medical issue I know quite a lot about, I've seen so much blatant false information from American doctors - but what they advise gives them the best chance of money from the insurance system. Cold but true.
So I've looked up the NICE guidelines and they have:
Common or very common
Diarrhoea; fever; gastrointestinal discomfort; headache; nausea; skin reactions; upper respiratory tract infection; vomiting
Uncommon
Akathisia; anxiety; arthralgia; asthenia; behaviour abnormal; depression; dizziness; drowsiness; dry mouth; haemorrhage; irritability; malaise; muscle complaints; oedema; seizure; sensation abnormal; sleep disorders
Rare or very rare
Angioedema; concentration impaired; disorientation; eosinophilic granulomatosis with polyangiitis; erythema nodosum; hallucination; hepatic disorders; memory loss; palpitations; psychiatric disorders; pulmonary eosinophilia; speech disorder; suicidal behaviours; tremor
Side-effects, further information
Eosinophilic granulomatosis with polyangiitis (Churg-Strauss syndrome) has occurred very rarely in association with the use of montelukast; in many of the reported cases the reaction followed the reduction or withdrawal of oral corticosteroid therapy. Prescribers should be alert to the development of eosinophilia, vasculitic rash, worsening pulmonary symptoms, cardiac complications, or peripheral neuropathy.
None of what you say is in the "common" (1 in 10 to 1 in 100) section. There are some in the "uncommon" (1 in 100 to 1 in 1000), which would worry me, but if you keep an eye out then you can probably mitigate the risk to a good amount.
The ones you are really concerned about fall in the "rare" (1 in 1000 to 1 in 10 000) or very rare (less common than 1 in 10 000) area.
I think I'd go back to the GP and ask if they can suggest something else because you're concerned about the side effects. They will probably try to reassure you.
Then you have a choice. You can start the course, it should be a low dose which should reduce the risk, and keep an eye out. Or you can refuse it, and may not get anything else. You have to decide the risk for your child.
eg. I wouldn't worry about it for dd1, because she would talk to me and doesn't have a tendency for any of those. I could also talk to her about the side effects without her worrying about if she had them.
I wouldn't touch it with a barge pole for dd2, who already has significant anxiety and has a tendency towards depression and dizziness. If I talked about side effects then I can guarantee she'd get 90% of them before the medicine had hit her stomach, and I can also say from experience that if depression hits, she masks very well until it's too much. It would be far too great a risk for her.
Ds I would consider. He has a nasty habit of when he is ill/gets side effects, he really is ill/has bad side effects. So that would make me nervous. However I could talk to him about the side effects and looking out for them, and I think he'd be fine (would probably tell me that he needs a cash rise in his allowance to mitigate them 😆).
I know if I gave him the choice, he'd probably choose not to take it, because he avoids medical intervention generally, but I would show him the side effects, point out the benefits, and say if he changes his mind (which he might do if the cough got too annoying) then to come back.