Parsing ふりする in song lyrics

Question

I recently started watching 宇宙よりも遠く場所 on Netflix and there's a catchy reoccurring insert song that gets stuck in my head. Looking at the lyrics, I'm confused by the second to last line:

波に揺られてかたふりして進んでゆくよ

To parse it, I tried adding some particles back in:
波に揺られていかたのふりをして進んでゆくよ

Giving me my guess of:
Pretending to be swayed (rocked?) by the waves, (we) move forward.

But I've never seen the かた (way of doing something) + ふりをする (pretending) structures used together before and it seems awkward somehow... Am I close?

Answer 1

I wonder if it's this かたふり：

「かたふり」とは船員用語で、船内で気が合う仲間同士で部屋に集まり、コーヒーやお酒を飲みながら自慢話やよもやま話に話を咲かせることを言います。話に熱中してくると身振りが大げさになり、自然と肩が振れるようになるという説や船の揺れに合わせて雑談をするからという説もありますが、真偽は不明です。

Translated:

かたふり is a word used by sailors. It means to gather with people you get along well with inside a room in a ship, drink coffee or alcohol, and boast or talk about various things. Some explanations of its origins are that when the talk gets heated, one's body's gestures become exaggerated and naturally one's shoulders (かた）will shake/swing (ふる) or that people talk while matching their movements to the shaking of the boat.

If it's this one, I guess this would translate to something like:

While being rocked by the waves, we get talk and get closer to each other on a boat and we move forward.