Not sure what's wrong with the interrail.eu (I never use that for reservations because of the extra fees they charge) but you can't book this train online on bahn.de. You can however book it via DB's phone service: +49 30 2970 (available 24/7). You can request to collect your reservations from a DB ticket machine, using the procedure called "Bahn-Tix".

Further, the only reason you're seeing this route via Mulhouse is because of the engineering works and bus replacement between Offenburg and Freiburg on that day. You can also avoid any obligatory reservations by using that bus service (it's non-stop). The arrival time in Luzern does not change. You can look up the timetable using "via Freiburg(Breisgau) Hbf” in DB's planner and use the "Seat only”/"Nur Sitzplatz” button to book any (optional) seats for 4 EUR per journey and person. Be sure to enter the complete journey because DB charges optional reservations per journey and not per train.

I've tried to link up with the company, but unfortunately without succes. Are you able to provide some more info? This is what comes up when I search with the "Nur Sitzplatz” option checked:

Without that option it does give me prices, but they seem pretty high considering we already have the tickets.

Thanks in regards :)

Please read my first reply: "you can't book this train online on bahn.de. You can however book it via DB's phone service: +49 30 2970 (available 24/7). You can request to collect your reservations from a DB ticket machine, using the procedure called "Bahn-Tix".”

Personally, I'd use the bus replacement as indicated by seewulf and book the ICEs via bahn.de for 4 EUR per person per journey.

So this should be it right?

Yep.

Thanks, we booked our reservations just now. We also tried to add the journey to our pass, but somehow it's different than the journey online.
 

In my browser I get this journey:

How do we get our journey in the app?

If you wait, maybe the app will get an update and you can add the journey from the app. If it hasn't changed before you travel, you can add the missing bits manually. There's always a link at the bottom of the planner results to do that.

As rvdborgt mentioned, you can add them manually through the blue hyperlink at the bottom of the screen. 

Thanks, we can use the manual additions as tickets whenever requested?

If your train does not require reservations and cannot be found in the app you can add the journey manually. Note, a journey is not the same as a seat reservation.