OK, I've done a little experiment that might help...
Gauge plate (aka ground flat stock) should work. Cheap and readily available in a wide range of thickness and sizes, plus it's normally magnetic. A piece the size you're after is probably about $50.
Typical flatness is within 50um, usually less (1/1000 inch, 25um), so it might be good enough as is. If it's not, then any machine shop with a surface grinder should be able to dress the surface for you. Alternatively just check the supplier's stated tollerances, you can find some flat to within 5um.
Thermal expansion of steel is about 9um per metre per degree C, so it shouldn't affect the surface flatness. (The width and length will change slightly, but not by much.)
As for the warpage, I've just clocked out a 250mm length of gauge plate on my CNC machine. It's 9.5mm thick but only 50mm wide (sorry, I couldn't find any nearer to 130mm at the required length). Laid flat on the bed, it shows no more than 40um deviation in thickness, which is within the spec.
I then supported it at both ends to measure the sag, and it's also no more than 40um difference along the the whole length. So doesn't look like it's really sagging at all over a 250mm span. If I press down on the centre I can increase the deflection by about 30um, but that takes quite a bit of force.
Just to be sure, I also quickly tested some 3mm plate, and it showed a sagging deflection of around 200um in the middle. (Wow! Interesting experiment!) So thinner plate definitely sags under it's own weight, which implies you need 8-10mm thick plate.
I've also weighed the 9.5x50x250mm plate and it's a tad over 1kg, so you're spot on at around 3kg. If you can live with that, then you have a possible solution.
For good measure I also checked for any vibration issues, but couldn't really measure anything. However if the light source is not rigidly attached to the plate, then it's something you might want to consider.