A simple fix is to add a very small, practically unnoticeable amount of monochromatic noise. For example, in PS, add a new layer above your existing image and use Filter > Noise > Add Noise... Use something like 3-5% (or even lower) and monochromatic. You can use a mask on this layer to add the noise only to the white area, so the image content does not get noise applied. The amount of noise you add does not need to be that great, just enough to get the JPEG compression scheme to see that the background is not a large area of contiguous, identical blocks of color.
Save the file with JPEG quality maximum ("12" in PS, or 100%).
As an example, I created a 5000x4000 px white document in PS (8bit, sRGB). I saved the document with JPEG max - the file size was 128 kB. I added 5% monochromatic gaussian noise and saved again - file size 11.4 MB. With 2% noise, I get a file size of 8 MB. Etc.
You can adjust the percentage of noise and the opacity of the noise layer to tweak the file size, as well as changing the JPEG compression quality. You can also apply a Gaussian blur to the noise layer to change (smooth) the distribution of discretely different pixels, essentially making the white area with noise more uniform - this will make JPEG compression more efficient and decrease the file size and break up any discrete noise that you may see at 100% zoom when viewing.
This should give you the flexibility to target the file size you need.
As a bonus, a small amount of noise, or "grain," sometimes improves the impression of image acutance - maybe your image will serendipitously get this little boost from adding noise too!
kirk