The current versions will likely work if I can figure out what files to edit to get Calamares installer to install with 1024MB RAM. We did some experimenting some time ago and even had this working with Plasma desktop. None the less it would be better to use something like ROME xfce. LXQt would be a good choice also but there is bug.
To Download something right now it would have to be before Lx 4.0 so maybe Lx 3.03 would work. If not then try Lx 2014.2. They can be found here.
Edit: It is unlikely BIOS would get this wrong since recognizing the system hardware is part of it’s job. But cat /proc/cpuinfo will tell for sure. Alternatively if you try to boot a 64 bit ‘Live’ ISO and that does not work try one that is 32 bit.
With ROME LXQt iso 1761 I tried with required RAM 1.0 and Calamares said that was not enough. Tried with required RAM set to 0.8 and installation was successful. System boots and seems to work OK in very brief test.
For LXQt at present user will want to use tiled windows because of bug 2816 until that is fixed.
But this test suggests this is workable on computer with 1024MB. Xfce4 desktop would be another one to consider for this type of application. The isos are available here. User could dual boot LXQt and xfce4!