Chemical methods to detect cell division
The most common indicator for the growth of any kind of cell (population) I know of is the amount of DNA in that cell. There are a number of (fluorescent) dyes that bind to DNA in a stoichiometric fashion and can therefore be used to determine the (relative) amount of DNA inside a cell (is hope that's what you mean with 'biochromatic indicator'). The most common dye used in this way is probably DAPI.

In order to determine which cells are dividing, you'll obviously need a device that can measure the fluorescence of each cell individually- so either a microscope or a flow cytometer. Because growing mammalian cells (it seems to me that's what you want to look at) multiply their full set of diploid chromosomes in the S-phase of the cell cycle, you can discriminate growing cells from others by their relative DNA amounts:


A non-growing cell has only the diploid chromosomes (1x relative DNA)
A growing cell will have either the doubled set of chromosomes (2x relative DNA) if it sompleteded S-phase or an intermediate amount (1-2x relative DNA) if it's currently in S-phase


Figure 1 of this (really old) paper also shows, how DNA amounts can indicate at which point of the cell cycle a cell is (which tells you if its growing) works.

A note on cell viability:

Staining any kind of cell with DAPI (or most other dyes that bind to DNA) will most likely kill it or at least act as a very strong mutagen. Based on your answer to my comment this should not be a problem.

While not really part of your question, you mentioned that you want to preserve cells using lyopreservation, which basically means drying them. You should consider, that mammalian cells will most likely not to survive this process (unlike plants and some invertebrates there are no mechanisms in mammalian cells to protect them against the water loss). Without sorting the live cells from the dead ones first, your measurements may detect cells that were dividing prior to the preservation.