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Well, there's a long history of dead bug construction, and 3D sculptures. But it's the short leads thing that's biting you here. You can solder the capacitor lead to the copperclad really close and use the capacitor as a sort of standoff sticking up. Then solder the lead to the inductor to the other cap lead really close. Extra length in the inductor isn't so bad. And, be careful about the inductor windings being close to the copper clad - the capacitance from the winding to the copper clad is essentially shunting. (The overall effect would be to lower the cutoff frequency of the filter).

This is actually pretty useful for you - the NanoVNA said "the filter's not working right", and as you change things, you'll see it improve. Which is a whole lot better than a sort of handwavey - lead lengths are important. You have a chance to quantitatively see how long a lead makes a difference.

BTW, people designing and building these filters often take the original design, build it up, find out it doesn't quite work, and then adjust the values of the components until it does. Maybe they change the 270 pF to 330pF (to account for the inductance in the leads). This is a lot of the "art" of building RF circuits with real, non-ideal components.

I think we've all been burned by something like this (resistors that are inductors is a notorious one)