I would think that it would actually serve to stablilize the cycle throughout the process. The chaeto would be able to absorb excess nitrogen compounds, minimizing swings in chemistry.
Yes, it will compete with bacteria for nutrients, limiting their growth. However, if you were to cycle your tank, then add chaeto, competition would still occurr and there would be some nominal swings as the chaeto and bacteria find an equilibrium. What will happen is, bacteria will starve and die, decompose, add nutrients to the system and feed the chaeto and the remaining bacterial population.
I would add it while cycling. Then, as always, add fish slowly, over time. It's all about balance and everything will balance out if given enough time.