Will Slick Tires Work for a Mountain Bike Ridden by a Heavier Rider?
Those tires should work just fine. Any of the MTB slick tires listed on that site would work well with your bike and current rims. 

You can get too small a tire on too wide a rim, but it takes a much bigger jump than from 2" to 1.5 or so. There is a very conservative sizing guide on Sheldon Brown's bike pages. 

Given your weight, I would try and find the largest size slick you can. Something like the Schwable Big Apple in a 2.0" format1 supports a maxim load of 275 lbs per wheel, but it's outside your price range, or from that page the Kenda K1029 Kwick Roller Sport 26 Inch Tire (50-559).

The last number is key to understanding tire sizes. You have 559mm rims, so just about any 559 tire will work with your bike. The 50 is the nominal width of the inflated tire in mm's. ( i.e. 50mm ~ 2 inches ).



1 Schwalbe has a sizing guide too.