The Apple 12" RGB has one video mode.
The LC has two video modes that work with the Apple 12" RGB display. They have the same horizontal and vertical timing but the width of the pixels changes such that the resolution can be 512 wide (square pixels) or 560 wide (not square pixels).
For 512x384 60Hz, the pixel clock is 15.6672 MHz and the total width is 640 pixels.
For 560x384 60Hz (Apple II modes with 192 doubled lines), the pixel clock is 17.234 MHz and the total width is 704 pixels.
In both cases, the total height is 407 lines, the vertical frequency is 60.15 Hz, and the horizontal frequency is 24.48 kHz. Does your LCD support that horizontal frequency?
The Apple 12" RGB can accept both modes because it doesn't know about pixel width - it's analog, there's no pixel clock transmitted to the display so you could put a million pixels on each line and it would still work.
So, how is an LCD supposed to know the difference between a 512 wide mode and a 560 wide mode? It doesn't. First, does your LCD work with the 512 wide mode? Then it should work with the 560 mode. But it might not sample all 560 pixels? Ideally, you want an LCD with analog input that samples the pixels at 17.234 MHz or greater, regardless of the number of vertical lines. Does that exist? You would need to verify by running some test patterns - or look at the text in 80 column mode, black and white mode.