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I've been looking for a bunch of different damage calculators but I can't find one that allows me to calculate damage using Bronzong's HA. Unfortunately I also have no idea how to do this by hand.

I can tell you how I'd do it using a little math first, but I'm not sure whether there's a calc for the entire thing.  Anyway, Heavy Metal always increases weight using the same simple function.  So you can figure out the weight after HA (and Sylveon's weight too), then use a Heavy Slam guide to figure out how much the base power is.  After that it's a standard power calc.
Ok so I just did the Heavy Slam Base Power calc for you.  Bronzong's final weight is over 18 times greater than Sylveon's weight. Bronzong would only need to be 5 times heavier to achieve maximum base power, which is 120.  Now if you put 120 base power into the online calc you're using it will give you the correct dmg.  I can't do this for you because it will need the attk power of your Bronzong and the defense of the Sylveon.  

FYI it's likely a OHKO... 120 base power stab supereffective rarely isn't a OHKO...

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Like they said in the comments, the base power is 120. Assuming we don't consider the effects of rounding, items, status conditions, stat changes, critical hits, any moves other than heavy slam, EVs, natures, and both Pokemon are level 100 and have perfect IVs, Bronzong's physical attack would be 214 and Sylveon's physical defense would be 166. The damage formula is ((2*level+10)*attack*base power)/(250*defense)+2. If we substitute all variables for their values, we get ((2*100+10)*214*120)/(250*166)+2. I got 131. If we then factor in STAB and the type chart, we get 393. Then, there's a random multiplier between 0.85 and 1 (full details here), so we do between 334 and 393 damage. (though it's probably be a few points off from a properly rounded calculation) If we're still ignoring EVs, Sylveon's max HP is 331, which happens to be just enough for a guaranteed OHKO.

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Thanks for the link. So if Bronzong's weight without Heavy Metal is already enough to provide a 120 BP Heavy Slam, then is there any point to the extra weight? The doubled weight doesn't further increase the BP so no further damage is inflicted right?
I know. It was asked in the question, so I answered it.