Overnight train Innsbruck to Frankfurt--Bayern Nacht?

Is there some specific reason you want to make that into an overnight trip? It's 5.5 hours and you can easily change in Munich. You can buy right from the Bahn website. Buy early - fares go up. I see really inexpensive fares for a few months out starting at 39 euros and they go up to 129 euros for trains this week.

I believe I saw an overnight train that was no stops and 8 hours, if I remember correctly--I thought that might be a good alternative to getting a hotel room in Innsbruck, since availability is virtually gone, especially for a family my size.
Whether I go ahead with the overnight train or not, could I still get the Bayern ticket even though the trip is originating in Innsbruck and not closer to the border?

I don't see a non-stop overnight option, or even a direct (no-change) overnight option. I just randomly chose the night of Friday, Dec. 9, for the departure, so I suppose there might be something different available on other days, and this itinerary might not be available on all other days. This is what I found:

1:28 AM: Dep Innsbruck on EN 467 (with sleepers)
3:23 AM: Arr Salzburg. 65-minute layover
4:28 AM: Dep Salzburg on EN 462 (with sleepers)
6:10 AM: Arr Munich. 42-minute layover
6:52 AM: Dep Munich on IC 728 (no sleepers)
10:04 AM: Arr Frankfurt

Try going back to wherever you found the itinerary you're interested in and click on "Show Details" or the equivalent. If you don't find something like that, look for a "+" sign. There should be something, because there's not going to be a non-stop train running from Innsbruck all the way to Frankfurt.

I looked for cheap flights on skyscanner.com but found nothing with a remotely decent price for the two random dates I checked well into the future, so it looks like a day train is probably your best option.

I'm sorry that I don't know whether the Bayern Ticket reaches as far as Innsbruck; if not, perhaps it works as far as Salzburg. There are others here who are very knowledgeable about German train deals, so I hope someone will be able to clarify that for you soon.

We rented an apartment here for Innsbruck: https://www.bedandbreakfast.eu/bed-and-breakfast/innsbruck/apartment-lisa-luise/327600/. We had the Luisa apartment, I think. They have large units and are about $200 per night for a group your size - didn't know your travel dates so I couldn't check availability.

(1) Is the Bayern Nacht ticket available when originating in Innsbruck? (2) Also, should I purhcase tickets on OBB or Bahn.de?

(1) No - Bayern tickets are valid inside Bavaria only. Moreover, those are regional travel tickets, which you cannot use on a long distance (night) train. There is one night train from Innsbruck to Düsseldorf, stopping at Frankfurt Südbahnhof (20:44 - 5:24; but what will you do at 5:30 in the morning in Frankfurt?).
(2) That makes no difference. Well, if saver tickets are sold out at one Website it may by worth while to double check at the other one.

the Bayern Ticket .... perhaps it works as far as Salzburg.

It does, but Salzburg is on a different line. On the both Innsbruck-Munich lines the BY ticket is valid from Kufstein or Mittenwald, resp.

I just randomly chose the night of Friday, Dec. 9, for the departure

That is not the best choice, because it is the last day of the 2016 schedule. On the weekend of Dec. 10/11 there is the yearly coordinated timetable change of all European railway companies.

Anyway, I would prefer a day connection from Innsbruck to Frankfurt, which is 5:30 hours with one lay-over.

That night train (EN 420) makes several stops throughout the night, including Munich, Augsburg, and Nuremberg, where it sits in the station for 1 1/2 hours.

I'd suggest forgetting the night train. For one thing, that's an easy trip to make in the daytime. If there are no hotel rooms available in Innsbruck, try Salzburg or even Munich for one night.

For rail trips in Germany, the easiest place to buy tickets is bahn.de or Trainline.eu.

If there are no hotel rooms available in Innsbruck, try Salzburg or even Munich for one night.

... or in one of the smaller towns of the Inn valley like Hall in Tirol. Or have an intermediate stop in lovely Mittenwald at the end of the Karwendel Bahn from Innsbruck, just over the border in Bavaria.

The Innsbruck-Frankfurt Sud-Dusseldorf night train number 420 starts service from Dec. 11, with the annual schedule change. It also stops at Frankfurt airport.

As mentioned above, the Bayern-Ticket is only valid for trains within Bavaria and to/from several "border stations", like Salzburg, Kufstein, and Ulm. The Bayern-Ticket Nacht is valid for overnight travel, Sunday thru Thursday from 6 pm to 6 am the following morning, Friday and Saturday until 7 am. However, it is only valid for regional trains running at night, not for night trains, which are long distance trains of the class IC/EC. Just like the Bayern-Ticket is valid for regional trains during the day, not for ICs, ECs, or ICEs.