Welding plastic with a heat gun
There are tools designed specifically for proper welding of plastics, for production of plastic water tanks, piping, and many other industrial purposes. Many of these tools are somewhat expensive and require compressed air or other gasses. I'm focusing instead on a cheap, hobby substitute for the proper tools.
This method works well for HDPE and PP plastics, which won't accept glue. If a type of plastic accepts glue well, such as polystyrene, acrylic or polycarbonate, use glue instead.
These plastics melt at a temperature around 500 degrees Farenheit. Needless to say, this will burn you. Wear gloves such as leather welding gloves, or at least heavy leather gardening gloves.
The first step is to create a pattern for the rim of the underside of the saw. I measured and cut a template out of cardboard, and made sure it fit the saw. Then I could start cutting the bucket to match the template.
I found that the best way to press the two parts of plastic together is between two pieces of sheet metal. The metal will not bond to the plastic after it has cooled for thirty seconds or so, and will lift right off.
The lesson from this is that when you modify or adapt a tool, be careful. Every power tool should be treated with respect as a potential source of danger, and some are more dangerous than others. It would have been easy to start a fire and perhaps burn the house down with a tool modification like this. Be cautious in experimenting, and have safety equiment like a fire extinguisher available.
Spot welding attempts
I tried heating through a piece of sheet metal with the soldering gun, but didn't have success. It spread the heat too broadly, melting the top tab but not the bottom tab.
The best result came from heating the head of a screw with the soldering gun until it sunk into the plastic, through the top tab down to face of the bottom tab. It did manage to melt the two tabs together, but only in a small ring around the head of the screw, with poor mechanical strength. Based on that, I gave up on spot welding and returned to the heat gun.
In the movie Toy Story 2, Stinky Pete appeared to be a nice guy but turned out to be trouble. In the world of common plastics, PETE is trouble too. PETE, polyethylene terephthalate, is the very common plastic used for all sorts of uses. Nearly all transparent food containers and bottles, such as the 2-liter soda bottle, are made of PETE.
Like the other polyethylene plastics, PETE cannot be glued satisfactorily with any readily available glue or solvent. It cannot be welded with a heat gun or soldering iron either. The thin walls of common PETE packaging will shrink and warp before melting with a heat gun. They will melt with the soldering gun, but don't seem to fuse to other melted pieces with any strength. Apparently PETE can be welded with industrial plastic welding equipment, but that is beyond the scope of my attempts. I've read that PETE can be glued with a hot-melt glue gun, and supposedly that's how most retail labels are attached to PETE containers. But I didn't have any success with it; the glue just peeled off. Contact cement and silicone sealer/caulking will hold slightly, but with poor strength, and will peel also.
The only success I have had with PETE has been with various forms of mechanical fastening. Screws and nuts will work if large washers are used to spread any stress. Rivets will work also. You can use epoxy glue through a hole, with a blob on each side. When it hardens it is essentially a rivet. One technique I like is to cut slots and tabs, like you would with a paper model. Fit the parts together, and then use clear packing tape on each side to hold the tabs in place. It works fairly well.