No.
Acid-base and respiratory physiology can be a little confusing.
If I took someone normally healthy, and injected a few hundred MEq of hydrochloric acid, and rapidly dropped the pH, we'd see a shift of the Henderson-Hasselbach equilibrium towards CO2, and an increase in PCO2, and (all other things being equal), a sharp increase in PETCO2, as you described.
H+ + HCO3- --> H2CO3 --> H2O + CO2
But metabolic acidosis doesn't generally develop like this, especially the DKA that you're describing. It occurs more gradually.
So as excess metabolic acids are produced, the pH drops and the concentration of hydrogen ions (or [H+], if you will) goes up. As we've said this pushes things to the right, and more CO2 to be produced (as the [HCO3-] decreases).
We'd expect if more CO2 is produced, the PCO2 and (all things being equal) the PETCO2 to go up. And if we had a paralysed, intubated patient, and we were controlling their respiratory rate, we'd see this.
But in a patient with intact respiratory drive, they're going to respond to the acidosis by increasing their tidal volume and respiratory rate. This is going to increase the elimination of CO2.
So our PaCO2 ends up being determined by the balance between CO2 production (increased by acidosis), and CO2 elimination (increased by hyperventilation / tachypnea). If the rate of CO2 elimination exceeds the rate of CO2 generation, then the PaCO2 will actually go down.
Even if PaCO2 is trending downwards, we may still be eliminating more CO2, as our minute volume is going up, compared to before the acidosis started.
[Note that a consequence of CO2 removal, is further pushing of the reaction towards the right, consuming [H+]. This is respiratory compensation. If we're lucky our respiratory system can compensate for the metabolic acidosis and keep pH >7.35. If we're unlucky, we decompensate, and pH falls.]
I hope this helps.
[Just an obligatory note that PETCO2 doesn't correlate with PCO2 under all circumstances. But let's not worry about that for the purposes of this example. ]