Fortunately I don't need to filter my water, tap water quality in my country is one of the best in the world (so much so that Kikkoman built their European soy sauce factory here in the Netherlands specifically because of the high water quality). I only filter water for my espresso machine to prevent scale buildup. Never tried boiling water for tea on my induction top, smallest size is 12cm (4.7") and the power setting delivers 3680W so it should be very quick, but still not as quick as just having instant boiling water on tap.
Instead of a mug, you could try a small induction compatible kettle or just get a cheap electric kettle. You can easily find a >3000W kettle for less than €50.
In my kitchen I have a tap that provides both hot and boiling water, to do this there's a 7 liter insulated vat of water kept at boiling temperatures at all times. Because it's well insulated it uses surprisingly little energy to keep it at temperature, about 10W (and between solar panels and battery storage, power usage isn't that much of an issue to begin with).
When you ask for normal hot water, it will mix the boiling water with cold water to provide instant hot water. You can also hook up a regular hot water line from your heater, that will make it switch to the regular hot water line once hot water arrives, but for regular kitchen use that's not even necessary so you can also run it without connecting it to hot water.
The boiling water tap is amazing. You can make tea directly from the tap. Want to cook some pasta, simply fill a pot with already boiling water, no need to wait. Also small stuff like adding some water to a simmering sauce that's getting too thick without it going off the boil. Quickly blanching some vegetables for stir-fry by simply putting them in a sieve and holding it under the tap. It seems like an unnecessary luxury but it's surprisingly useful.
According to Wiki itâ(TM)s the spin-off of IBMâ(TM)s infrastructure services devision.
[Crash programs] fail because they are based on the theory that, with nine women pregnant, you can get a baby a month. -- Wernher von Braun