In case of Mittenwald-Switzerland journey you actually have a few options:
a) cheapest one will be to purchase in advance discounted tickets from Scharnitz (first Austrian station after the border) to Buchs(SG) via OeBB website. If you do so early, you have a chance to buy tickets for as low as 9 euro, which is a super bargain in this case
While Mittenwald is just 6 km from Scharnitz and even full fare ticket is really cheap. Or alternatively you can even take a walk over the border
b) other option is also relatively cheap and what is important doesn't require any prepurchase of tickets allowing you to be flexible. Idea is to travel to Switzerland via an opposite way, thus not through Innsbruck, but with 29 euro Bavaria Lander ticket (valid for up to 5 people) from Mittenwald to Lindau, while then just a full fare ticket till first Swiss station - St. Margrethen (straight forward international Lindau-St.Margrethen will cost 7,4 euro per person or instead domestic Austrian ticket Lochau-Hörbranz-Lustenau, thus covering segment from the German-Austrian border till Austria-Swiss border for 3,6 euro per person). Alternatively you can also forget about transiting Austria at all and just continue from Lindau or neighboring Wasserburg by
ferry to Swiss Rorschach (should be a nice journey over the Bodensee)
c) if again think about going via Innsbruck, then in case you've already missed discounted tickets till Buchs it might make sense to buy discounted domestic ticket till Bregenz (also from 9 euro and a bit easier to catch), while from Bregenz it's 15 min. ride to St.Margrethen)
d) another option without necessary ticket prepurchase is just buy regular Austrian domestic tickets from Scharnitz with Minimax group discount (-40% off regular fare makes it 101 euro for 4 passengers). Obviously it's almost three time more expensive comparing to four 9 euro tickets, but you stay totally flexible and can buy such ticket even 5 min. before departure.
p.s. just in order to compare: full fare (=when you purchase at the station before departure) Mittenwald-Innsbruck is 11,1 euro per person, Innsburck-Buchs (SG) you already know, while direct international ticket Mittenwald-Buchs via Innsbruck is around 40 euro per person