I suspect the core of your confusion comes from not knowing a meaning of 筋{すじ} which is applicable in both of your sentences; that is:
❺ 物事の論理的な流れ。道理。
「➖の通った話」
明鏡国語辞典 第二版
This usage of 筋 is pretty idiomatic and mostly found in the expressions 筋が通る or 筋を通す, which you should be able to look up directly.
「〜するのが筋だ」 stems from that usage and basically means that 〜 is the reasonable, logical, right, or natural thing to do.
The ってもんだ is best explained in detail elsewhere (see this answer, h/t Chocolate for finding that), but it’s basically a masculine-ish way of giving a “that’s the way things are” sort of finality to your sentence. Overall, I would say that 筋ってもんだ is not so common a collocation to be able to treat it as a single unit, so it is likely better to consider them separately.