Deutschland ticket

I'm pretty sure that, in the same way that the Bayern Ticket covers only that portion of the bus #840 route which travels through Germany, the D-Ticket will not cover Salzburg > German/Austrian border on this route.

To use ONLY the Deutschland-ticket for this trip, travel by train only from Salzburg to Berchtesgaden via Freilassing.

but can you go from Salzburg to Berchtesgaden on the #840 bus using the Deutschland ticket?

No, and with the train that only works because due to an approx. 150 year old treaty between Austria and the Kingdom of Bavaria Salzburg was defined as a common border station. This does not apply to the bus lines that were added much later. Cf. https://www.dbregiobus-bayern.de/angebot/freizeitbusse/rvo-rva/watzmann-express (in German, last paragraph; use google translator or DeepL)

The rules for using the Bayern-Ticket on the Watzmann Express (RVO 840) are pretty well defined in the tariff section of the RVO website (www.rvo-bus.de), and I would expect it to be the same for the Deutschland Ticket. The Bayern-Ticket does not cover segments of RVO bus lines that are in other countries. So, it should be possible to pay the bus driver for the segment from Salzburg (multiple stops in town including the Hbf and Mirabell Palace) to the Austrian/German border (near Schellenberg). Then the Deutschland Ticket would cover travel from the border to Berchtesbaden Hbf. That would only amount to a few euro.

I have also read posts here that the bus driver just passed on charging for the Salzburg-Schellenberg segment and accepted the Bayern-Ticket for the entire trip, but I don't think you could count on it.

Or, as mentioned above, just go by train from the Salzburg Hbf to Berchtesgaden. That way takes about 25 min longer and requires a change at Freilassing Hbf, but you'll know for sure it's covered by the Deutschland Ticket and you'll travel in a larger train coach rather than a more restricting bus. If you are coming back to Salzburg from Berchtesgaden, the last bus leaves Berchtesgaden at 6:15 pm. while the train runs until after 10 pm.

The rules for using the Bayern-Ticket on the Watzmann Express (RVO 840) are pretty well defined in the tariff section of the RVO website (www.rvo-bus.de) [...] The Bayern-Ticket does not cover segments of RVO bus lines that are in other countries

It would be nice if it were so crystal clear. The website of the Austrian cooperation partner Albus explicitly says that the Bayern Ticket is valid on the entire route:
https://albus.at/de/linie-840-watzmannexpress-2020/

But no conclusions can be drawn from this for the Deutschlandticket, because this would have required a separate contract, which did not materialize.