Let's go through it, bit by bit.
「年長者」 can mean "older person", but in context it can also mean "the oldest person within a given group". 「AでありB」means "~is A as well as B". So this part means that person is an older person (or the oldest person of the possible candidates?) who also happens to be a doctor. There is no actual indication of seniority as a doctor in the original sentence.
「武闘」 means "combat" and 「派」 refers to groups of people who are separated by given differences(i.e 猫派vs犬派). So in this context, 「武闘派」 likely refers to people who don't really like to cooperate with other people and instead are very aggressive in trying to get their way. 「Aとは一線を画する」 means to draw a clear line between A and oneself, so to put it all together, this means that the person being described has drawn a clear line between himself and the aggressive people at the Ministry of Science and Technology.
「敢えて」 refers to doing something that doesn't appear like a good idea or is difficult to do with some kind of intent. This makes it clear that 林 becoming the 実行者 does not appear to be a logical choice, or that there were other candidates for that position that seemed more likely to be appointed.
So to put it all together, a more accurate translation would be something closer to
The reason why Hayashi, who is older, a doctor, and has distanced himself from the aggressives at the Ministry of Science and Technology, was purposefully chosen to carry out (the project? the mission?) is unclear...
The 「が」 at the end is difficult to translate without more context. It can either be a "While" to be placed at the very beginning of the sentence, or be translated as a "but..." at the very end.