Safe yet strong exothermic reaction?
Here are some more options:


Pd-catalyzed oxidation of dilute hydrogen gas in air: $\ce{2H2 + O2 -&gt; 2H2O}$  If you use a dilute hydrogen gas mix (say 95% nitrogen and 5% hydrogen), no flaming combustion can occur in air.  But oxidation still happens, at least when catalyzed by palladium particles. This is how microbiological anaerobic chambers maintain a zero-oxygen environment.  You can probably buy some catalyst fairly cheaply from a microbiology anaerobic chamber company like Coy labs.
Phosphorus pentoxide hydration.  Wikipedia says this reaction, $\ce{P4O10 + 6 H2O -&gt; 4 H3PO4}$, has $\Delta H_{reac}=-177\text{ kJ/mol}$.  There is no flame and no fumes, but the reaction does create phosphoric acid, which is corrosive and can be dangerous.
A modification of your sulfite / bleach idea: use sodium dithionite instead of sodium sulfite.  Sodium dithionite ($\ce{Na2S2O4}$) is air-unstable but if you can work out a way to handle without air exposure it will be more exothermic than sulfite.  Instead of dithionite itself you could use the much easier to handle Rongalite and hypochlorite.  Possibly, using calcium hypochlorite suspensions in water (mostly insoluble) will allow you to use higher concentrations of hypochlorite and Rongalite without the reaction rate and temperature skyrocketing (since the rate will be controlled by $\ce{Ca(OCl)2}$ dissolution).