score:8
The western half of what you're looking at would fall well within the range usually referred to as Allgäu - here is a 'map' of the outlines of that region as used in today's tourism industry, covering both Germany and Austria although originally Allgäu was a part of the German Oberschwaben, see here.
The Austrian areas are part of (Nord)Tirol - a map with names of parts of Tirol is here - and sub-regions of the Austrian Tirol are found here.
The German areas are all part of the German region of Oberbayern although that stretches much further north than you want to look. Looking at the Landkreise in that region will be quite tedious, but gives fine grain control over where you're looking at. These are the Landkreise bordering Austria in that region (a map is here):
Upvote:3
The bit you have shaded is not a defined region in itself, as Greyshade has already pointed out. Being a local, I would split the German part into three areas, from West to East:
There would be no combined name for the area, since it is composed of two distinctly different regions with different dialects and different customs that do not see each other as belonging together: The Allgäu region belongs to the Swabian dialect area and the Pfaffenwinkel is the South-Easternmost corner of the Bavarian dialect region within Germany. You will notice a rather sharp change in the locals’ speech once you cross the Lech river.
The downside of searching for something in the Allgäu — especially the Ostallgäu, eastern Allgäu — is that it does extend a fair bit further North, away from the alps. If you are looking for something within the alpine region, i.e. close to the mountains, try Allgäuer Alpen (Allgäu Alps).
The bit of Austria you marked also is rather interesting. While nominally beloinging to Tyrol it is typically referred to as the Außerfern — it is außerhalb (behind) the Fernpass when viewed from the rest of Tyrol. Interestingly, the dialect in the Außerfern is also of the Swabian type and it is the only bit of Austria that features Swabian. (Vorarlbergian is an Alemannic dialect, not a Swabian one.)
In any case my hat is off for going on holiday in this incredibly beautiful region, no matter whether it is the Allgäu, the Außerfern or the Pfaffenwinkel.