I think if I were doing that trip, I would base myself in Berchtesgaden or Salzburg first, with a day trip to the other, then move to Oberammergau for the three Schlösser. Myself, I would prefer to stay in the much prettier Berchtesgaden, but Salzburg is a little faster to get to, and either way there is the RT bus trip to the other. Then I would move to Oberammergau. O'gau is a short bus trip to both Hohenschwangau and Linderhof and not to far from Munich for a day trip to the Oktoberfest. All this travel can be done by public transportation.
If you insist on a single base, then it would depend on whether you are using just public transportation or a car. Using the trains, Munich is right in the middle between Füssen and Salzburg, and you don't want to use it as a base for 5 days, so I would pick someplace west or east of Munich on the route between them.
On the west, I would use Landsberg am Lech as a base. From Landsberg, it is about 1h40 to Füssen, with a change of trains in Kaufering. Kaufering, actually, is on the main line from Munich to Füssen, about 25 minutes from Landsberg on a spur line. Landsberg is a delightful place to stay in it's own right. I stayed here - it's economical and convenient. I was there the last weekend of Oktoberfest, and it was busy, but not booked.
From Landsberg, it's about 45 minutes by train to Munich Hbf, so you could do the Oktoberfest as a day trip. The route to Salzburg goes through Munich, so you could stop there on the way to Salzburg.
Landsberg is closer to Füssen than it is to Salzburg. On the east side, Rosenheim is a major train stop and closer to Salzburg.
If going by car, I would choose somewhere in the Voralpen south of Munich. The quickest and recommended car route from Füssen to Salzburg goes through Bad Tölz, so that is what I would pick as a single base with a car. You could use the rail line Bayerische Oberlandbahn (BOB) to get into Munich so as not to have to park there.