You are completely correct, T1 diabetic folks who exercise, do and should run higher than the textbook "target" at the gym to prevent lows thereby. Consider this, to get an A1C of 8 you have to be running higher than you realized consistantly.
A single reading of 200 something at the gym is NOT going to skew your overall reading meaningfully. The rest of the day you aimed for (hit?) your target, only at the gym, you aimed higher. After the gym you play catch-up, when its necessary. A normal karate class lowers me a hundred points very consistantly regardless of the particular exercises.
Hope you're approach works for you, think your doctor is ~playing games~ with you though. In the real world, real diabetics sometimes go higher for a perfectly valid purpose. And no ~checking~ during is necessary.