Sheldon's chart is very conservative. Keep in mind that the higher the pressure you run the greater problems you may have. This is why MTBers get away with 2.1" tires on narrow rims.
You can certainy try 32mm tires on your bike as long as you accept some risk. Just start with low pressure and no load. Ride the bike around town casually - don't bomb any uber steep hills. As you get confident you can increase the pressure and add some cargo. Do this incrementally.
If you are totally risk adverse - ignore what I said above and get new wheels built that accommodate your desired tire width.
As a data point for comparison of the 8 bikes in my apartment 7 have tires that are too wide for their rims according to Sheldon's chart. I haven't had any issues at all, but I do run wider tires at moderate pressure as my experience has shown that you don't get anything but a harsh ride when you inflate tires past a certain point.
I'll agree that there is a difference due to pressure, however you can run very wide mountain bike tires at very high pressures (60 psi) for their width. Even at low pressure, mountain bike tires go through more abuse than road tires do with greater changes in tire volume than road bike tires every will (decrease the volume of a tire and the pressure will increase)
I got out my handy calipers and went out to measure my rims and tires. Both mountain bike have 25mm rims (external width) with 48 mm and 53mm tires on them. My road bike has 26 mm rims with 32mm tires on them. My touring bike has 24mm rims with 37 mm tires. Both of these bikes have relatively low pressure tires (90 psi) mounted on them but you could certainly mount higher pressure tires on them without issue.
I measure the internal width of a rim that is the same width as the others and came up with measurement of 18mm so the walls are 3mm each or 6 mm total. Given that measurement, I'd put sunburst's rim width as around 19 mm. That should easily take a 32 or even a 35 mm tire without problems.