I went through what you are going through currently - both when I was on imatinib (which ultimately failed to work for me) and later dasatinib.
My neutrophils went as low as 0.1 - which is dangerous. I was told to go immediately to the hospital if I developed any fever whatsoever. At 0.1, you can die from what would otherwise be a routine infection. This is why it is very important to have weekly CBC blood tests during early phase of TKI treatment until blood counts stabilize.
What is happening is your leukemic blood system (which is working, just poorly when in chronic phase) is being replaced with new normal blood. Imatinib is killing off the CML cells and leaving a void sort of speak for normal hematopoiesis to re-populate your neutrophils. Couple this with the fact that most TKI's will also suppress normal WBC production to a degree and you have dropping neutrophil counts.
Your doctor is doing the correct protocol by stopping treatment and letting your blood system recover. This also allows your leukemic system to recover somewhat as well, but with each successive re-start of therapy and stopping, your normal blood will get the upper hand and the leukemic system will get wiped out (except for quiescent leukemic stem cells).
It took me 4 stop/start cycles before my low neutrophil count stabilized and started to rise up to normal while remaining on my TKI (dasatinib).
There is a paper (I would have to find again) describing statistically those who suffer myelosuppression with imatinib will also suffer myelosuppression with dasatinib, but will have a much deeper response overall and with a much lower dose. If you are switched to dasatinib, consider re-starting on a very low dose (20 mg) and track your response. This happened with me. I re-started on 20 mg dasatinib and within a year or so achieved PCRU and am now drug free (TFR).
You'll get there. Be vigilant and ask lots of questions.