Yes. In FM some of the power shifts into the sidebands. At a modulation index around 2.41 (e.g. 1kHz baseband, 2.41kHz deviation) the carrier disappears, then at higher values of m it goes antiphase. You would get a greater effect at very high mod index, as you suggested (e.g. 100Hz baseband, 75kHz deviation gives m=750). This is because Bessel functions oscillate but get smaller at higher values. See
Wikipedia, and the article on Bessel functions it links to. You could skip the Bessel maths and just look at the graph of the first few Bessel functions J0, J1 etc. The carrier amplitude is given by J0, first sideband by J1 etc.
I don't know what the minimum bandwidth of your analyser will be but if it is sufficiently narrow you could see this effect.