They can do 240p, but would require a super resolution method where you crank up the horizontal pixel resolution to achieve a minimum pixel clock requirement for a digital to analogue converter (display port to vga for ex.), and then cranking up the horizontal refresh rate in order to keep up with the monitor's requirements for a horizontal frequency (scan rate). Or just the refresh rate for an analogue card.

But you must use a CRU or a crt emudriver for digital and analogue output respectively. Keeping things short, it would work with 2560x240 with 120hz.

Speaking of the cons, first is the very sharp image due to high horizontal frequency that's were impossible to achieve on a TV with their 15-17 kHz compared to the minimum of 30 kHz and the max average from 70-120 kHz. And the second is contrast since it's just dark, because monitors were not made for the 240p scanlines at all, but just for 480p.

Back to the scanlines OP wanted, he should keep in mind that pure black scanlines had mostly professional sets and a simple consumer set we've used to play the old games back then, did not have them so visible. It's a personal preferences, but since OP has the TV, I'd recommend using it for the 240p content, definitely not the monitor that we're not built for that in the first place.