Confirmed. Trying to upgrade a weapon with not enough money ends up with dialog box with “not enough money” text. But with enough you can upgrade weapon for free.
For some reason upgrade to lvl 8 costs me 300 rupee(i dont remember what was required money amount). At this point i think i saved ~1 zeny using broken upgrade system. Which is decent amount of money. If i had to farm 1 zeny on top of what i did farm already i think the game experience would be completely different. At least i had to have certain amount of money per weapon level to not waste weapon experience. Which is completely unique task.