Honestly, if you want to do justice to your Sikkim Tour, dedicate all 7 days to Sikkim alone, cover west Sikkim and north Sikkim also (besides Gangtok etc) which are amazing (I understand that there is a direct flight from Delhi to Gangtok now as Gangtok Airport is operational, pl check up). I don’t foresee any issue with your 2 Y old (just keep him adequately warm during journey and when outdoor).
Lachen You could start with
Buddhist monastery Tour to rhododendron sanctuary Visit Gurudongmar lake
Explore unesco site Kanchenjunga park. Shopping is popular here along with some Sikkim cuisines. Since you are travelling with a child, it’s advisable to avoid adventure sports but if either of you are interested trekking and wildlife sightseeing is popular. Yumthang
You could start with phensang monastery
Visit seven sisters falls
Drive to Chungtang to watch the conflux of rivers
Explore a day in Chopta valley. Darjeeling Nightingale park is a popular spot
Darjeeling peace pagoda
Lamahatta eco park
Shopping For adventure trekking nature camping is popular. Travelling with a small baby has many do’s and don’ts. Have a thorough checklist before travelling. Kindly carry required medicines and all baby items for travel. There are some best hotels safe enough to try Sikkim cuisine kindly check them out before travel. With a full plan you and your family are safe enough to travel. Happy safe journey.
The tour and travel companies can cost you a lot instead you can connect with local people. to make permission is not difficult you can make it from Mangan also if you have all the papers with you, but you do need to take care of drivers permit too and his fees. As for the child I would not suggest you to go with a 2 year old to lauchun, yumthang and Gurudongmar as its risky. The climatic conditions are not that great and people have breathing issues. so for kid I wouldn't recommend. But you can go to Nathula instead of that. And I must say you should go Pelling, the west side it is absolutely amazing. no permissions needed not that bad conditions to breathe, kanchanjunga is easily visible. From there can go further to yuksom, kheechuperi, Dubi monastery. Uttarey is also a good village to be. You will find cheap home stays here with great food. I can help you with few contacts there you can text me on insta - @theunfoldingplay