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.

That said, the volume of water sitting in the pipes between the water heater and the bathroom faucet is a major factor in how long it takes the hot water to first reach the tap. The key aspects are;
How well insulated is every inch of the water pipe all the way back from the back of the faucet to the tank outlet?

How direct is the pipe run from tank to faucet?

What diameter is the water piping?

This last one may be a surprising factor. The smaller the pipe diameter the faster the hot water 'front' travels through the pipe. Not only is the overall volume of water inside the small pipe less, but the smaller inside diameter of the pipe reduces the mixing of cold and warm water as the hot water pushes the cold water forward.

Faster water velocity inside the pipe reduces mixing and delivers hot water faster. This must be balanced against pressure loss with very long pipe runs and high flow showers and bathtub fillers.

With a typical 1/2" copper pipe, the hot and cold water inside the 'hot' pipe mix as you run the tap, while the temperature front inside the pipe slowly progresses forward. Eventually the water coming out begins to warm up, but it still takes a while longer for the fully hot water to arrive. The metal mass of the copper pipe itself also draws some warmth from the hot water, further delaying the arrival of truly hot water.

This is one of the reasons a PEX flex tube from a faucet back to the central water distribution manifold will typically be a small inside diameter (3/8" perhaps). There is less mass of standing water inside a smaller tube to lose heat, hence less energy loss overnight. Faster water velocity inside the PEX means quicker arrival of fresh hot water when the tap is first turned on in the morning. The PEX tubing has a lower thermal mass so it does not cool the hot water flow as much.

The small inside PEX pipe diameter is not a problem for a low flow bathroom faucet, the pressure loss is modest. Since the PEX itself is flexible there can be fewer or preferably no right angle elbows so water pressure loss through the smaller diameter flex tube is similar to a larger size traditional rigid copper pipe installation. For a bathtub or shower of course a larger diameter PEX tube would be used.

Depending on your particular structure and where the existing pipes are, you might be able to add a separate flex PEX tube directly from the faucet back to the hot water tank. Properly insulate that thin flex tube and you should have rapid hot water arrival in the morning, and almost instantly even a few hours later.

A different option is to install a hot water circulation pump. These use a temp sensor and a small electric pump to circulate the hot water through the existing hot water pipe to underneath the faucet.The cooled off water is sent back to the water heater to eventually be reheated.

When you turn on the tap the hot water is always right there, inches from the faucet.

Here is an article discussing such things. Waiting for Hot Water

This one is more in depth. Hot Water Distribution Research

And this.