Tiaral

In the third option, everything is very simple regarding skirmishes. This can be justified:
1) By the fact that we have sworn allegiance to a particular country. Therefore, it is our duty as a vassal.
2) By the fact that we are not vassals, but allies of one of the parties in the conflict. Military assistance is part of the agreement.

Added 1 minute ago
Participation in the voyage can be justified by an interest in exploring new lands. Well, in the case of the third option, again. Although I'm not sure if it's worth sending a leader there.

No, look. Every order had a goal that was formulated briefly and comprehensively. For example, "liberating the Holy Land from pagans" or "searching for the Holy Grail," or, if the order was founded by some king or his sycophant, "protecting the royal family." On our side, there is also such a goal - "ensuring the well-being and success of _insert_name_here_", but we cannot clearly declare it because it is, well, eee, not respectable. That is, the options you list provide answers in a specific situation, but the goal should be formulated in such a way that there is no need to look for a special answer in each situation.

Added 2 minutes ago

The Most Honourable Order of the Bath, more commonly known as the Order of the Bath[1], is a British order of chivalry founded by George I on May 18, 1725.

Okay, we've decided on the name, now we need to determine the goal.