Van 't Hoff equation
The T that is held constant on the LHS of your equation has a different meaning from the T that is allowed to vary on the RHS.

The van 't Hoff equation answers the following:

Suppose I have a reaction that takes place at const T, p at T = T1.  Now I change to a new temperature, T2, and allow that reaction to again take place at const T, p, except now T = T2.  How does the equilibrium constant change?

Now that you understand this, you can see there is no inconsistency between "constant T" and "changing T", since they refer to two different things.

I.e., the fact that the reaction is constrained to take place at constant T in no way prevents me from testing the reaction at a different, but still constant, T.

The analogy I'd give is this.  Suppose we want to determine the effect of altitude on 100 m sprint times.  The 100 m is always run on a level surface (no change in altitude during the race).  Yet this does not in any way prevent us from comparing 100 m run times at different altitudes (e.g,  sea level vs. 10,000 feet).

So your question is like asking:  "How can we determine the effect of changing altitude on 100 m run times, given that the 100 m run is always done at constant altitude?"  Well, we can, of course.