the TMC389 should not become very hot itself - unless you have much additional load on the +5V output. But, it is important to have the exposed pad under the IC soldered and contacted to your board GND plane with a number of wires, in order to dissipate heat. Then it will heat up by 15 to 30°C, depending on the supply voltage.
I have looked up your MOSFETs: The N-Type IRFZ44N has a quite high gate charge, which makes slopes slow, and is non-optimum with 5V gate drive. When I check IR types, IRLZ44ZS or IRLR3105 or IRLR2905Z or similar would be better (all 55V). The P-Type has a quite high resistance, IRFZ9Z34 would have less. If you have the chance to purchase them, SUD23N06 and SUD08P06 are a very good choice.
In case, you do not need 48V supply, there are MOSFETs with better RDSon and less Qg, e.g. 30V types or 40V types. Also, a FDD8424H will improve power dissipation by a factor of 3, I guess. We found it is good for 5 to 6A, also! (do you have the TMC389 Evaluation board? - you could test by swapping sense resistors).
For reducing standby current, you need to make SW interaction. You could use the TMC429 position reached interrupt, to detect stand still of motor. Then you need to write to TMC389 a reduced current setting. When you start the next move, you need to increase the current again before start.
The coolStep function in the SMARTEN register will not help for motor stand still. It will help optimizing power at a medium velocity (not too low, not too high), when your motor runs for a long time at this velocity (e.g. at VMAX during a normal move).