Introduction of an asset tax in the AK model
I think your math is mostly correct but I have to admit that I am not used to AK models.
A short answer for your main question:  is it ok to introduce taxation in the model without including the government?
Answer: Yes, of course. There are overwhelming literature on dynamic or static theoretical modeling of the effect of tax policy without introducing government as an agent.
For a most famous example, see J.E. Stiglitz, Effects of wealth, income, and capital gains taxation on risk taking. Quarterly Journal of Economics, 83 (1969)
In this line of research, people almost never introduce government. You could search for the works by Arrow (1965), Cass (1972), Fishburn, Richter (1960), Tobin, Mossin (1968), Merton (1969), Russell and Smith (1970)
Also you could see the first paper published by Acemoglu.
But, as you might already explored, many recent papers include government as an agent. For example: Farhi, Emmanuel. &quot;Capital taxation and ownership when markets are incomplete.&quot; Journal of Political Economy 118.5 (2010): 908-948.