If anything, "Prussia" would still be founded in a way, only fanatically Catholic and with no delusions of being anything other than an army with a state.
It would be interesting to see the evolution of a surviving crusader state into the Renaissance. Teutonic relations with Poland, Lithuania, and Novgorod will almost certainly continue to be terrible. Expect even more tension and on-and-off wars than OTL, as the order will continue to seek to expand its influence, politically and militarily, at the expense of the Slavs and Balts.
As for what role the order will play in German politics? It will most likely continue to receive patronage from the Holy Roman Emperor and other well-off nobles, even from non-German christian states, as it did OTL. The biggest wildcard is the Papacy, which in OTL attempted to seize all Order assets for "misconduct" (read: ethnic cleansing) against the devoutly christian Kingdom of Poland in 1309. If the Papacy is able to convince the HRE to cut ties with the crusader state once it becomes too powerful, the order will be diplomatically isolated, especially once crusader crimes against the christian Slavs and Balts become more well-known following Tannenburg.
Yet at the same time, even with the lack of foreign support, a victory at Tannenburg would increase the strength of the order exponentially, to the point that it may very well stand on its own just as easily as it would with Imperial support. Even with Papal objection to the more seedy doings of the order, the christian foes it would potentially face are already fighting it anyways, as the HRE would not seek to destroy a useful buffer state.
But would the order seek to influence "secular" politics in the neighboring Empire? I would think so, as the statelets of the HRE are its primary source of manpower for the short term. It will never seek to directly challenge the power of the Emperor, and will be ineligible to join as a proper member as it technically isn't a monarchy. But its influence on the German states will be undeniable, potentially influencing how the Protestant reformation plays out, if at all. Perhaps they end up as a sort of proto-Landsknecht loyal only to the Emperor and "true" Christendom, acting as the sword and shield of the HRE (and potentially serving as the linchpin for a true imperial army down the line). Meanwhile, only Emperors in the pocket of the order will be elected through the mobilization of vast sums of funds and influence, serving to act as its mouthpiece-by-proxy.