Is current due to a point charge moving in a circle ill-defined?

There are two aspects to the question though second is complementary to the first point: what is the time scale of our experiment? And how are we going to use the current defined in our problem?

Before that if our current is discontinuous or very wierd kind of function from analysis there is no need to get anxious it tells us about either breakdown of our laws, ex singularities of GR, otherwise how we modelled the physical problem, ex the periodic Dirac Delta potential while giving elementary calculation of band structure: there are Delta everywhere and it's periodic after large N.

First, if $t_{exp}>>T$ then we have the steady state current given by your formuale, if this seems outrageous two examples may help iron coil voltmeter behaves like direct current of $I_{rms}$ is being sent through it even though we have a oscillating current which varies in milliseconds. Magnetisation of materials uses above formulae and it does give good experiment result, though the explanation actually comes from QM. This regime is magnetostatics.

On the other hand if $t_{exp}<<T$. The charge looks stationary to the experimentalist and one is free to use electrostatics.

At last if $t_{exp}approx T$. The regime is electrodynamics. And you have to use Jefimenko equation for a point charge all sorts of things are thrown in retarded time, Lienard-Wiechert potential factor. See $10.67$ Griffiths. And you get physical implication of radiation.

Now onto the next point how are going to use our defined current? It depends on our experiment precision. Just a simple example will suffice if we had the condition of $t_{exp}>>T$ then the generalised result $10.67$ reduces to Biot-Savart rule with current substituted by your formulae but lets say you did the experiment you find discrepancy from the approximate formulae then it will lead you to the investigation of much better approximation.

TL;DR $I = Sigma_t q delta (t-nT)$ where n is an integer here $delta$ has dimension of time inverse hence correct dimension of current. Just for sake of completion $$I_{avg}=frac{1}{t_{exp}}int_0^{t_{exp}} dtSigma_t q delta (t-nT)$$ if $t_{exp}>>T$ $$I_{avg}=frac{1}{t_{exp}}mq$$ where $m=t_{exp}/T$ since $t_{exp}>>T$ so $mpm1$ doesn't change much. Hence we get the usual formulae $$I_{avg}=frac{q}{T}$$