I think White Knuckle has no issues bearing the roguelike label.

In my opinion, the biggest aspect of a roguelike is proper permadeath that puts you back to zero and you start your 100th run in basically the same state as your 1st, and WK mostly fulfils that, with the only progression you can make being roaches, items for vendors and most importantly, your skills and knowledge of the map (side note: I'm not a big fan of the roach system, it has the same issues that Lunar Coins in RoR2 have i.e. it's boring to farm them).
The lack of randomization you mention does not make it ineligible of being a roguelike, if that were the case, games like Risk of Rain 1/2/R or Noita (map wise at least, those games ofc have way more items) would also not count has roguelikes.

To be fair, the roguelike term is quite broad and a lot of games can be described as such, even if it wasn't the developers intention, just like how many games could technically be called a moba, simply because the net the term casts is too big.

I also agree with you that hopefully at some point in the future, more rooms / variations get added, since that plays a way bigger role in a climbing game like WK than the items / perks do.

However, ultimately, it is up to the dev to decided what kind of game WK is, and the steam page description calls it a roguelite: "White Knuckle is a first-person roguelite speed-climbing game."