Travelling by train from Fussen -> Salzburg, use DB Nav, or DB Nav and OBB?

The 9 EUR ticket is valid from Fuessen to Salzburg as long as you use the BRB local train from Munich to Salzburg. Click on "local transport only" to see these.

Seems if I go via Munich I have to transfer from HBF to Munich Ost.

No, the regional trains to Salzburg start out at the München (Munich) Hbf. Their first stop is at München Ost. You'll transfer at München Hbf.

Of course, if you really wanted to, you could get off your train from Füssen when it terminates in the Hbf and take the S-Bahn from the Hbf to Ost, and catch the regional train there. But why?

BTW, June 12, 2022, is a Sunday.

There seems to be something going on with the tracks between München Ost and Salzburg. It looks like starting on Sunday (but not on Saturday, June 11) the Bahn is routing you from Ost to Salzburg on a more northern route via Mühldorf. The train from München to Mühldorf starts at Ost, not the Hbf. That's why they are telling you to make the transfer to Ost.

The Bahn says to get off the train from Füssen at Pasing and take the S-Bahn from there to Ost, but that train also goes on to the Hbf, getting into the "Starnberger Wing Bahnhof" on the north side of the Hbf seven minutes later. If I remember, there are stairs from the wing bahnhof down into the Hbf (tief) S-Bahn station. Seems that would be an easier transfer to the same S-Bahn you would have caught in Pasing after a platform change. That S-Bahn comes through the Hbf 23 minutes after the train arrives giving you plenty of time to catch that S-Bahn and a very real chance of getting an earlier one to Ost.

The Bahn is also suggesting a somewhat longer route that starts at the Hbf, but goes to Landshut, where you transfers to a train to Salzburg via Mühldorf.

It looks like starting on Sunday (but not on Saturday, June 11) the Bahn is routing you from Ost to Salzburg on a more northern route via Mühldorf.

No. Please note that there is a presently glitch on the side of the BRB (which runs the direct trains between Munich and Salzburg) that prevents the display of BRB trains in the DB app. There are in fact trackworks; nontheless most BRB trains start from Munich Hbf. The replacement time table is here:
https://www.brb.de/de/fahrplaene/linienuebersicht-fahrplaene/linie/re-5

Scroll down to

Chiemgau-Inntal: Fahrplanabweichung München - Salzburg 16. - 29.06.2022

and download the PDF.

I have no idea what is going on between Rosenheim and Freilassing, but there doesn't seem to be any regional rail traffic on that line, even between any of the stops. It's all by bus. The only regional rail connection I can see is from Traunstein to Freilassing, and that connection goes way to the north, through Garching (Alz) and doesn't use the mainline between those towns. Teisendorf is between those towns, but the only transportation between Teisendorf and Traunstein or Freilassing is by bus.

However, it does look like the long distance trains, at least Westbahn and RailJets, are going through on those lines. That often times happens, even when they are doing track work. They interrupt their work and put things back together long enough for the long distance trains to go through, but close the tracks to regional trains.

It looks like this interruption will only last until the following weekend. I wonder if they are just examining all the rails between Rosenheim and Freilassing because of the derailment of that train coming out of Garmisch-Partenkirchen on Friday.

but there doesn't seem to be any regional rail traffic on that line

So please look up the BRB time table for 06/12 linked above. There is the usual hourly BRB train with slightly longer travel times (leaving Munich Hbf at minute 44 instaed at 56).

As stated above, BRB trains cannot be displayed in the DB App at the moment (and until June 18) due to a technical glitch.

Thanks for all the replies.

The BRB app shows two options: the RE 5 or Zug 971. What's a Zug, is that eligible for the 9euro ticket?

RE 5 is the hourly BRB train from Munich Hbf to Salzburg. Zug is just a German word for train. I see a Westbahn 971 at 15:48. DB website says (in German) "not valid with the 9 EUR ticket".

RE # is the line number, Zug # is the number of the train.