Actually, the reason loyalty was so heavily stressed during the Sengoku-jidai and beyond was because of its scarcity. Lesser lords frequently betrayed their liege-lords in order to switch sides and maximize benefits for themselves. The 'loyalty propaganda' (for lack of a better phrase) came from the top down. Powerful Daimyo would stress the importance of loyalty and the evils of betrayal (employing and mobilizing Confucian doctrine) as a means of solidifying their powerbase.

To return to your question then, a Daimyo betraying another Daimyo was never a particular concern during the Sengoku Jidai (it would have been a bigger deal during the Muromachi period or the Tokugawa-jidai) because this was literally "The Warring States Era" (aka 'sen 戦 - war' 'koku 国 - country' 'jidai 時代 - era'). Each Daimyo was essentially the leader of an independent and self-interested country. It was perfectly reasonable for them to betray and attack one another during this period.

The larger concern was whether their vassals would switch sides and betray them during one of the constant conflicts happening between and amongst the various Han (domains). However, this practice in itself, although it was shunned by the Confucian morals in place in Japanese society, was never truly checked by these morals. It was too commonplace to stop. Not to mention the fact that if a vassal switched to the winning side, no one would bring up his betrayal (as he was likely of major assistance in the victory) and if he switched to the losing side he was likely executed (in which case he never needed to reconcile or justify this betrayal, he was simply punished for it [admittedly a harsh punishment haha] and society moved on).

It was not until peace reigned throughout Japan during the Tokugawa-jidai that this culture of loyalty truly solidified. During this period warfare was MUCH less common as central control by the Tokugawa bakufu forced loyalty into the rival Daimyo. Once the country was stable it became important to keep it that way and those who were disloyal were appropriately punished and shunned. It is this period where the Hagakure was written and the ideology of Confucian loyalty became fixed within the Japanese political system.

To conclude then, the Sengoku-jidai itself was too chaotic for anyone to truly be concerned with loyalty. A Daimyo was just as likely to encourage his rivals' vassals to betray the rival, thus flouting the "culture of loyalty" himself as he was to pontificate about the importance of loyalty amongst his own vassals. Power was all that truly mattered during the Sengoku-jidai and this culture of loyalty never really stabilized until the Tokugawa Ieyasu finally stabilized the country in the mid 17th century.

This 'culture of loyalty' did not really exist in Japan until the Tokugawa Era. It was brought up and hoped for during the Sengoku Era, but did not become a reality until unification had taken place.

Source:

Wm. Theodore De Bary - Sources of East Asian Tradition Volume 1

Hagakure - Yamamoto Tsunetomo