Such an unsolvable problem! Except that in many cultures it's been solved for a long time. You have two last names, one patrilineal, one matrilineal.
Hyphenated names go way back, and it's all about prestige. If the wife came from a prestigious family (too), they wanted to let the world know that these kids, they have connection to two prestige families. An example from my part of the world is the former defense minister Kristin Krohn-Devold. Other well known hyphenated prestige names in Norway are Rieber-Mohn, Tybring-Gjedde.
There's not an ever expanding list of hyphenations, that's an entirely imagined problem, and a bit of an "old man rants at cloud for not following tradition" thing. It's more common that the less prestigious name is quietly dropped, and you're back at an unhyphenated name. A fictional, but very plausible example is the main character of the Danish drama "Matador", who changes his name from Mads Skjern-Andersen to just Mads Skjern. I have many examples in my own family tree, of various distant cousins who got a prestige name of a non-paternal ancestor (it may be further back than the mom!) as middle name, and then just quietly dropped their patronym/last name so that the middle name was suddenly a last name.