If you have a triac type dimmer from Home-Depot, you could unsolder the wimpy one and replace it with the large one you have. And of course, heat sink it.
It depends on how the dimmer is designed. A lot of times the load is in series with all the other circuitry and you won't have a convenient way to get the trigger pulse. As was suggested, if the circuit board looks like it can handle the extra current maybe you can swap triacs and add heat sinking. Then hope the diac and your triac are compatible.
I actually tried it... The series connected Leviton brand dimmer (cheapest on the face of the earth) did not work to trigger the triac except in the high ranges.
Substituting the triac with the larger one did work better - but not smoothly I had to turn it up to ~15% to get it to turn on then could turn it down to ~5% where it would turn off if I tried to go lower (and that point was variable so it might turn off arbitrarily)
I eventually used a picaxe to read a pot and use pwm to control a solid state relay, and a tricolor led to indicate which stove range element was doing what from across the room... Cost ~$20 and a couple of days to convert two of the controls. New, so called, "infinite range controls" were ~$25 and only last couple of years the way I use the cookstove.
I expected the potentiometer to fail given the environment but they are still working after 4 years. The original thermo-mechanical controls had a time constant of ~36 seconds for the pwm, the electronic ones ~6 seconds.
I made a mistake on my schematic the first time through, the dimmer should connect the gate to MT2 (not MT1) the corrected schematic is attached below
Built it as shown, and it worked perfectly, it dimmed from very low power proportionally through to full power. I had it jury rigged, with just masking take covering the connections, and no heatsink on the triac, which was ok for brief testing, but would have overheated eventually.
The heater "sang" quite a bit since I used no RFI suppression caps or coils ... may change that, may not bother ...
I guess I'll build it into a box for permanent use.
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