I had tested Inkscape with an image in a similar size here and had no stress either in export time or in memory usage (although my system has more RAM than that, a crude monitoring I set up during export didn't show any significant memory usage when creating the output image).
What is confusing in Inkscape are the entries set-up in the export bitmap dialog - and I think maybe you are getting trapped into those. The varios x0, x1, y0, y1, width and height entries are relative to the internal resolution of the SVG file. And the "dpi" settings there correlates this internal resolution with the intended resolution of your output.
So, try just to set-up the desired output size in pixels in the Width and Height fields in the
Bitmap Size sections - put 10000 for the width, let it calculate the height automatically, and ignore its calculations for the DPI (they will show a ridiculously high number).
There is your exported PNG - without much stress to the system. The DPI resolution is a simple number set in the file that can be changed with another program (for example, using GIMP if you set the parameters in image->print resolution it does not change one single pixel in your image, just the internal metadata informing its size - and you can also re-export it to TIFF or PDF from there).
Actually if inkscape really fails, you could also try opening your SVG in GIMP, setting the DPI at import time so that it would have the desired size (upscaling it to 10000px wide inside GIMP won't work, since the program converts the image to raster data at input time).