I have wondered similar things before about how the Monastic Order of the Teutonic Knights could live, as a viable entity, far, far longer. But perhaps it's better exploring the "how could it happen" before you explore the "what would happen if". Reason being, in my opinion, the former would dictate the character of the latter considerably.
My own thoughts are that a decisive Teutonic Order victory against the Lithuanians during the siege of Vilnius (1390?) could allow this, but the order required many internal reforms before this to create a more durable system of governance. The shape and size of these could allow such an event but of course would have other impacts.
A similar issue would be what to do with the Livonian Order to the north. If memory serves, these guys not only created problems for the Teutonic Order but, through their autonomy, occasionally and deliberately worked against the Order itself (not in the pitched battle sense but in the stubborn, unconstructive sense). Subjugating this region and creating a far more centralised Order would have helped but again, how and at what cost?
Finally, there's also the issue of materiel from Germany proper (who supplied many men, Knights, gold etc. to the Order), so if you can find a way to lessen reliance on Germany (however that might manifest itself, e.g. expansion of the Order into Germany - but obviously not through conquest, as this is a Catholic Organisation that relies on the Pope for legitimacy, amongst other things) then you will be able to achieve a stronger state of affairs pre-siege of Vilnius.
Having said all that I am dubiously far from being an expert on this issue (I just read a really great book not long ago called "The Teutonic Knights: A Military History" by William Urban, which I thought was excellent - an expert may feel differently - nevertheless, I strongly recommend it, even if you don't go through with this TL

).
I'm not sure if this has been done before but it would be lovely to see it.