Hi Luckyrabbit:
In my area we still see all kinds of homes using 1.5" for their washing machine and utility sink drains and they are having very few issues...
Although 2" is ideal, you should be fine piping up the sink 1.5" and then installing a separate 1.5" laundry trap with 30" standpipe as you would like to do.
Just be sure that both fixtures get
properly vented.
I have attached an image showing how to pipe this up so the sink gets a vent and the washing machine drain also gets vented properly....fixtures are in the other room so you see the back side here.
If you can't connect the new laundry drain pipe back into its own vent then you may be able to use an air admittance valve (
AAV...see picture) to get this done! These must remain in the open or can be installed behind an access panel if necessary. If you use this for the sink drain it can be installed under the cabinet.
Let us know if you need more information...
MARK