The big water heater should not 'fire up' when you draw off 15 seconds of hot water unless the water temperature inside was already on the cusp of being cold enough to trigger the next reheat cycle.
The big water heater is also an on-demand (tankless) type, it always fires up when the hot water runs.
So why did you say it seems like a 'waste' for it to fire up when you are drawing hot water, since that is what is was made to do?
The argument in favor of tankless water heating is to (in part) avoid the parasitic heat loss from a hot water storage tank, trading the reduced standing heat loss from not having a storage tank for the (hopefully) sufficiently efficient tankless heating system.
If the tankless system is combustion powered then the energy cost delta between a compact electric storage heater and a fuel burning tankless becomes an operating cost differentiator.
My comments regarding timely delivery of the hot water to a bathroom faucet through the piping system should apply similarly for tankless and traditional tanked hot water sources.
One option I forgot to mention is installing an inline tankless electric heater just for the bathroom faucet, located in the cabinet directly below. Might require a dedicated electric cable feed to power it.
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