I found this paper on OSA: Control of residual amplitude modulation in Lithium Niobate phase modulators. It suggests a control mechanism which does not require controlling the temperature. The main point in it is that if a DC offset is present in the RF signal going into the EOM, it is effectively the same as changing the temperature by some amount. So, in principle, by changing the DC bias of the RF signal with a bias-tee just before the EOM, one can minimize the RF AM modulation. The paper though doesn't suggest the feedback mechanism or what sensor would be good for this. If we think this is worth giving a shot, in my opinion, if we fork the output of EOM somehow, measure the RFAM through it and employ a feedback circuit to control the DC offset, we can actively suppress the RFAM. Just a thought at this moment.

Just after writing above, I realized people at JILA have done something similar (with more complexity though):
Reduction of residual amplitude modulation to 1 × 10^-6 for frequency modulation and laser stabilization