Any idea how much the train pass is in Freiburg?
If you refer to the KONUS Card which offers free travel from Gengenbach + other towns, it is not available in Freiburg, not at any price. It is only available if you stay in one of the participating towns, all of which are listed on the link I provided previously. Here it is again:
https://www.dreisamtal.de/_Resources/Persistent/ee8509e131aa5b4f53135d921f3338c37a357c1b/7697983_KONUS_Flyer_GB_2023-Internet.pdf
Instead of booking so far south, I'd look around closer to Gengenbach. I did that myself (without dates) at booking.com on your behalf and found options for 5 persons in Offenburg, just a few miles from Gengenbach.
I also checked out some Black Forest tourism websites. Haslach - a very nice town just 12 minutes from Gengenbach on the scenic Black Forest Railway - turned up at the official tourist office page of listings. Haslach is very close to the places you said you wished to visit - closer than Gengenbach - and it's also a KONUS town. It looks like an option that might work for you quite nicely:
https://www.schwarzwald-kinzigtal.info/kinzigtal/ukv/house/Haus-zum-Hobel-Haslach-im-Kinzigtal-FIT00020070044085093
This place has two apartments - the "Zirbe" (which translates to Stone Pine or Swiss Pine) apartment has 3 bedrooms and accommodates 6. Washing machine included. Services include pick up from the train station (if you want it, for a fee I imagine) as well as bike rentals. The site has multiple photos and you can translate the text using your browser for further information. Minimum stay is 4 nights. The availability calender for August is nearly wide open. The location is convenient to the town center and the railway station on foot. Haslach is only 26 train minutes from Triberg, and also near the Black Forest Open Air Museum in Gutach.** Take the train to Hausach (5 minutes) and walk to the Museum and to the summer bobsled.
The homepage: https://www.haus-zum-hobel.de/ferienwohnung-zirbe/
This page, at a commercial site I am familiar with, has reviews you can translate:
https://www.traum-ferienwohnungen.de/51817/#bewertungen
I'd use the email adddress at the "Kontakt" at the home page to make an inquiry or a reservation. I am guessing there is no deposit required since the only payment method mentioned is cash. This is exactly the old-school-Black-Forest way that I have booked previously in the Black Forest at small private inns and apartments - send your personal contact information via email and request reservation for the dates you want, no deposit, show up with cash in hand for the amount agreed upon in the email. Such reservations are not necessarily refundable - German law is "if you book, you pay", unless a specific cancellation/refund policy is in place. So I would be sure to ask about their policy on this in my inquiry.
When we were in Haslach, the traditional costume museum was closed. But we did make it to the WW II work-camp memorial site. There is also a mine just outside town that you can visit.