LC filter applicability for power to output cables
An LC low pass filter can work effectively but, be aware of the new resonance that it brings to the party. That resonance would be at a lower frequency than the 240 MHz you are trying to eradicate but, nonetheless it is still there and if something reacts in a way that stimulates that resonant frequency, then you may solve one problem but create another. You need to damp that new resonance with resistance to prevent it from attaining a high Q factor and causing secondary problems. For instance, you can put resistance in series with the coil: -

Or you can put resistance in parallel with the capacitor: -

You can also put resistance in parallel with the inductance and the whole point here is that you are trying to eradicate 240 MHz, whilst not creating a resonant point that is too sharp.
Ferrite beads are naturally a turn-to choice because they are quite lossy at high frequencies and therefore act as a combined L and R but they still need some capacitance to ground to be fully effective at closing-down the frequencies you are trying to get rid of. You have to pick the right one of course such as this one: -

As you can see it has high losses at around 240 MHz but this is series loss and requires a capacitor down to ground to form an effective RC circuit. However, if C is chosen to be too big, a resonant frequency can be formed in the low MHz that is just as &quot;peaky&quot; as a conventional RLC circuit. For instance, at 2 MHz, the reactance is about 30 ohms implying a series inductance of about 2.4 uH and, if you used 2 nF as the output capacitor, you will create a big (and potentially problematic) resonance at 2 MHz because the resistive losses are ONLY about 5 ohms or lower: -

Pictures from this Interactive RLC page.
It should also be noted that the basic resonant problem doesn't easily solve itself by raising the capacitance to higher values because, at lower resonant frequencies, the ferrite bead is firmly acting as an inductor with very low losses and it isn't until the capacitance rises to values above 1 uF that the resonant peak is starting to be kept under control (circa 100 kHz).
If using a ferrite bead, add a series resistance of a few ohms or look into adding a resistor in parallel with the FB but, be aware that a parallel resistor will let more noise through at 240 MHz! Be also aware that above the peak frequency where the FB is most effective, its impedance turns capacitive so, at frequencies over 1 GHz, it may not be that useful. Choose carefully!

I am wanting to reduce the clock noise getting onto the supply line and going out the cables

Clock noise can have very many higher order harmonics so, if your basic clock is 240 MHz, it could still be producing a lot of noise well-into the GHz range. Given that a particular FB will turn capacitive above a certain frequency, in order to adequately get rid of clock interference on the line, it may be necessary to create a more convoluted solution using two or more different ferrite beads. Simulators are your friend and I would urge you to use them.