No, you can’t travel via Switzerland. 

try a search from Strasbourg to St Gervais Les Bains, via Cluses. Then from St Gervais to Chamonix.

(You may need to pick another city in France to force it to route you away from Switzerland.)

Since you won't be able to go via Switzerland (unless you purchase a Global Pass) the easiest route is as often in France via Paris. It looks like a big detour on a map but all high-speed railways lead to Paris. You need to transfer from Gare de Lyon to Gare de l'Est so it's a bit annoying, have at least 1h margin for the connection. Change in Bellegarde (and sometimes Annemasse) afterwards.

If you want me to have a look let me your travel date. For example on winter weekends + holidays there are direct TGVs from Paris Gare de Lyon to Annemasse/Evian-les-Bains.

This is a good route which avoids Paris and Switzerland :

- TGV Strasbourg - Mâcon 09:05 - 12:18 10-20€

- TER Mâcon - Bourg-en-Bresse 13:04 - 13:34

- TGV Bourg-en-Bresse - Bellegarde 14:12 - 15:00 10-20€

- go drink a coffee at the station or something

- TER Bellegarde - Annemasse 16:15 - 16:55

- TER Annemasse - St-Gervais-les-Bains 17:02 - 18:16

- TER St-Gervais-les-Bains - Chamonix 18:28 - 19:13

A bit of an adventure on hidden lines, esp. Mâcon - Bourg-en-Bresse !

Alternatively, you may wish to purchase a Saver Day Pass / or a ticket to allow you to travel via Switzerland.

Because the 5 days One-country Pass costs 221$ and the Global Pass 317$ I agree that purchasing a Saver Day Pass for Switzerland makes more sense.

It costs 52 CHF full-fare bought in advance. Valid on all public transportation of the country, from Basel SBB to Châtelard-Frontière on your itinerary.

It still makes sense to buy a 5-day pass and not a 4-day one for Strasbourg - Basel SBB + Châtelard-Frontière - Chamonix. If you decide to go via Switzerland you save at least 1 but more likely 2 TGV reservations too.