Currently window size isn't visible to Hatari, it's SDL internal thing. Hatari just sets internal framebuffer size and SDL2 scales that, to whatever size it deems suitable.
Looking at the SDL2 documentation, there seem to be SDL_GetWindowSize() and SDL_SetWindowSize() functions which might be used for implementing something like that. Saved Windows size should probably be ignored if user specifies on command line any of the numerous options that might affect the window size (machine type, TOS resolution, zoom/aspect/etc options), which makes it a bit more complicated.
I may take a look at it at some point, but probably not in next few weeks. Ping in this thread if nothing happens by end of May.
PS. Regarding the earlier question on larger window size... On Linux, with SDL1 there's a way to get Hatari window content show up larger also in ST mode. One just needs to have a display mode set for X server, which matches closer to the Hatari window size (e.g. 640x480 if Hatari borders are disabled and using 2x zooming option for low-res). Normally I would expect X to handle that automatically though. Supported resolutions can be seen in "xrandr" output.