There are two websites which will help you all the while in Switzerland.
One is the timetable:
Please read the instructions carefully. There is a lot of information buried in the layers of menus.
The other is a map with transport information:
Here is Champoussin;
Under "Points of Interest" in the menu, click on Traffic which will allow you to put an overlay on the map showing public transport stations.
You have picked a remote village with limited public transport. It looks to me like the main reason for its continued existence is skiing in the winter...several ski lifts are nearby. Many older remote mountain villages have depopulated unless they have such a source of income. This one looks like it has modern construction. Looking at the map in detail, with rows of identical buildings suggest that it might be a purpose built village of apartments or chalets, built as vacation home or apartments for skiers. Some of the street names are clearly modern, including Rue Kappa and Rue Gamma.
It is built on the side of a hill:
map.geo.admin.ch/?topic=ech&lang=en&bgLa yer=ch.swisstopo.pixel karte-farbe&layers=ch.bav.ha ltestellen-oev,ch.swisstopo.swiss tlm3d-wanderwege&X=117584.00 &Y=555550.00&zoom=10
The trip will involve trains as far as Val-d'Illiez, then a bus. There is limited bus service from Val-d'Illiez to Champoussin. On weekdays it has the typical commuter pattern - two buses in the AM at 0713 and 0920 and two late in the day at 1620 and 1820 to the village.
The first part f the train trip from Zürich to Lausanne, is on fast mainline trains. It would be necessary to check the timetable to see if the route to Val-d'Illiez from Luzern would take less time than traveling from Zürich . I think it might; just guessing. But,it is not certain; the time table would tell you. Routes would probably merge at Bern...and travel time from Zürich or Luzern to Bern is similar.
Departures from Champoussin are: 0637, 0750, 1040, 1540 and 1740. You can use the timetable to figure out how to get to Geneva.
Bus service on weekends to Champousin is likely to be on a different schedule. .I did not check because I think you would be traveling on the 5th and 8th.
Use the timetable to estimate times to, for example, Titlis. Engelberg is the base for Titlis:
Travel from Champoussin to Titlis will be limited by the bus service from Champoussin.
I have attached a screen grab from the time table for the 8th for a trip to Engelberg.
It will take 5 to 6 hours to get to Engelberg. It may not be possible to return on the same day in time to catch the last bus up to Champoussin, even if you leave Engelberg as soon as you arrive, skipping Mt. Titlis.
You will note that the route requires that you go to Luzern, then to Engelberg.
Possibly, you could organize that in some way if you stayed in Luzern, rather than Zürich
You would have to work the timetable carefully, and study the Titlis page linked above to refine that idea.
Depending on your flight time back to India, you might enjoy Montreux more than Geneva. there is excellent train service from Montreux to Geneva airport.
At Montreux, there is a very nice alpine railroad to Rochers-de-Naye, which could be a substitute for Titlis.
That might be possible as a day trip from Champoussin. Check the timetable.
Unless you plan significant hiking, which seems unlikely unless your child is able to hike, there is probably not much to do in Champoussin in September.
August has more events.