About the file size difference: the jpeg stores less image data AND it is compressed (even if not much by being set to highest quality).
About keeping the same dynamic range: you can't at all if saving as jpeg. Hue and luminance info will get set to the nearest set bandwidths.
You might have a better go if you use a tiff with a close bit value to your raws, ie, a 16-bit tiff rather than an 8-bit tiff, the difference
being that it is lossless compared to the data loss due to compression in a jpeg.
And then: what's the color depth of your monitor? - And the program you're using to view it with?
But if you want: to show a fair representation of the tonal (and colour) values of the raw file then go along the lines that Nardes says.